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Properties

Label 8-45e8-1.1-c1e4-0-5
Degree $8$
Conductor $1.682\times 10^{13}$
Sign $1$
Analytic cond. $68361.0$
Root an. cond. $4.02115$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 7-s − 8-s − 11-s + 2·13-s − 2·14-s + 16-s + 11·17-s + 2·19-s − 2·22-s + 15·23-s + 4·26-s + 29-s − 4·31-s − 2·32-s + 22·34-s − 37-s + 4·38-s − 5·41-s − 10·43-s + 30·46-s + 20·47-s − 15·49-s + 20·53-s + 56-s + 2·58-s + 17·59-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.377·7-s − 0.353·8-s − 0.301·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s + 2.66·17-s + 0.458·19-s − 0.426·22-s + 3.12·23-s + 0.784·26-s + 0.185·29-s − 0.718·31-s − 0.353·32-s + 3.77·34-s − 0.164·37-s + 0.648·38-s − 0.780·41-s − 1.52·43-s + 4.42·46-s + 2.91·47-s − 2.14·49-s + 2.74·53-s + 0.133·56-s + 0.262·58-s + 2.21·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(68361.0\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{16} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.809435194\)
\(L(\frac12)\) \(\approx\) \(9.809435194\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + p^{2} T^{2} - 7 T^{3} + 11 T^{4} - 7 p T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + T + 16 T^{2} - 3 T^{3} + 117 T^{4} - 3 p T^{5} + 16 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + T + 19 T^{2} + 74 T^{3} + 167 T^{4} + 74 p T^{5} + 19 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 22 T^{2} - 73 T^{3} + 341 T^{4} - 73 p T^{5} + 22 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 88 T^{2} - 451 T^{3} + 2111 T^{4} - 451 p T^{5} + 88 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 49 T^{2} - 34 T^{3} + 1115 T^{4} - 34 p T^{5} + 49 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 149 T^{2} - 1026 T^{3} + 5553 T^{4} - 1026 p T^{5} + 149 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - T + 76 T^{2} + 55 T^{3} + 2597 T^{4} + 55 p T^{5} + 76 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 82 T^{2} + 345 T^{3} + 3405 T^{4} + 345 p T^{5} + 82 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + T + 49 T^{2} - 392 T^{3} + 241 T^{4} - 392 p T^{5} + 49 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 139 T^{2} + 454 T^{3} + 7829 T^{4} + 454 p T^{5} + 139 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 124 T^{2} + 704 T^{3} + 6293 T^{4} + 704 p T^{5} + 124 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 295 T^{2} - 2890 T^{3} + 22931 T^{4} - 2890 p T^{5} + 295 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 298 T^{2} - 3001 T^{3} + 25499 T^{4} - 3001 p T^{5} + 298 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 17 T + 238 T^{2} - 2095 T^{3} + 18809 T^{4} - 2095 p T^{5} + 238 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 241 T^{2} + 2288 T^{3} + 21959 T^{4} + 2288 p T^{5} + 241 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 17 T + 304 T^{2} - 3117 T^{3} + 32001 T^{4} - 3117 p T^{5} + 304 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 244 T^{2} + 1441 T^{3} + 24947 T^{4} + 1441 p T^{5} + 244 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 196 T^{2} - 679 T^{3} + 18071 T^{4} - 679 p T^{5} + 196 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 283 T^{2} + 1590 T^{3} + 32439 T^{4} + 1590 p T^{5} + 283 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 30 T + 620 T^{2} - 8415 T^{3} + 89871 T^{4} - 8415 p T^{5} + 620 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 257 T^{2} + 1998 T^{3} + 31929 T^{4} + 1998 p T^{5} + 257 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 19 T + 469 T^{2} + 5368 T^{3} + 71215 T^{4} + 5368 p T^{5} + 469 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.46906113105488149454462881169, −6.33894003377905974872338788770, −5.80541952820827538846547857482, −5.75948377190534970946833370950, −5.50678061296201193626064401357, −5.21219836113836705714892453675, −5.18940974835223129927983744077, −5.17552455575905478013320325446, −5.04664841668109489348325199681, −4.52876502312444769094391772939, −4.29671585207682245253933946503, −4.08327006510256462781802320264, −4.03444666176582085483552494455, −3.56560251152700243403749272035, −3.37903516643879692097624704174, −3.30360244554730699438722768821, −3.22732172636275883827214305887, −2.70638916567970191410781593872, −2.51779283108621967671914315955, −2.30332829111001375439013057095, −1.70440772349425881315142198940, −1.43521928269557906204356746021, −1.09166969958562837572328242616, −0.886399801437427913467559339949, −0.47398112452799819696800086071, 0.47398112452799819696800086071, 0.886399801437427913467559339949, 1.09166969958562837572328242616, 1.43521928269557906204356746021, 1.70440772349425881315142198940, 2.30332829111001375439013057095, 2.51779283108621967671914315955, 2.70638916567970191410781593872, 3.22732172636275883827214305887, 3.30360244554730699438722768821, 3.37903516643879692097624704174, 3.56560251152700243403749272035, 4.03444666176582085483552494455, 4.08327006510256462781802320264, 4.29671585207682245253933946503, 4.52876502312444769094391772939, 5.04664841668109489348325199681, 5.17552455575905478013320325446, 5.18940974835223129927983744077, 5.21219836113836705714892453675, 5.50678061296201193626064401357, 5.75948377190534970946833370950, 5.80541952820827538846547857482, 6.33894003377905974872338788770, 6.46906113105488149454462881169

Graph of the $Z$-function along the critical line