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Properties

Label 8-1617e4-1.1-c1e4-0-5
Degree $8$
Conductor $6.837\times 10^{12}$
Sign $1$
Analytic cond. $27793.8$
Root an. cond. $3.59330$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·3-s − 4-s − 8·5-s − 8·6-s + 6·8-s + 10·9-s + 16·10-s + 4·11-s − 4·12-s − 8·13-s − 32·15-s − 6·16-s − 20·18-s − 8·19-s + 8·20-s − 8·22-s + 24·24-s + 28·25-s + 16·26-s + 20·27-s − 16·29-s + 64·30-s − 16·31-s − 2·32-s + 16·33-s − 10·36-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.30·3-s − 1/2·4-s − 3.57·5-s − 3.26·6-s + 2.12·8-s + 10/3·9-s + 5.05·10-s + 1.20·11-s − 1.15·12-s − 2.21·13-s − 8.26·15-s − 3/2·16-s − 4.71·18-s − 1.83·19-s + 1.78·20-s − 1.70·22-s + 4.89·24-s + 28/5·25-s + 3.13·26-s + 3.84·27-s − 2.97·29-s + 11.6·30-s − 2.87·31-s − 0.353·32-s + 2.78·33-s − 5/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(27793.8\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 7^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr C_2\wr C_2$ \( 1 + p T + 5 T^{2} + 3 p T^{3} + 11 T^{4} + 3 p^{2} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 36 T^{2} + 116 T^{3} + 293 T^{4} + 116 p T^{5} + 36 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 50 T^{2} + 240 T^{3} + 915 T^{4} + 240 p T^{5} + 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 46 T^{2} - 48 T^{3} + 958 T^{4} - 48 p T^{5} + 46 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 58 T^{2} + 192 T^{3} + 971 T^{4} + 192 p T^{5} + 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 38 T^{2} + 200 T^{3} + 494 T^{4} + 200 p T^{5} + 38 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 180 T^{2} + 1372 T^{3} + 8501 T^{4} + 1372 p T^{5} + 180 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 190 T^{2} + 1464 T^{3} + 9470 T^{4} + 1464 p T^{5} + 190 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 66 T^{2} - 112 T^{3} + 1227 T^{4} - 112 p T^{5} + 66 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 60 T^{2} + 436 T^{3} + 3410 T^{4} + 436 p T^{5} + 60 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 158 T^{2} - 1432 T^{3} + 11262 T^{4} - 1432 p T^{5} + 158 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 36 T + 660 T^{2} + 7756 T^{3} + 63197 T^{4} + 7756 p T^{5} + 660 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 270 T^{2} + 2376 T^{3} + 22070 T^{4} + 2376 p T^{5} + 270 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 148 T^{2} - 180 T^{3} + 10701 T^{4} - 180 p T^{5} + 148 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 156 T^{2} - 1048 T^{3} + 13782 T^{4} - 1048 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 194 T^{2} - 1256 T^{3} + 7947 T^{4} - 1256 p T^{5} + 194 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 372 T^{2} + 3948 T^{3} + 574 p T^{4} + 3948 p T^{5} + 372 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 98 T^{2} + 96 T^{3} + 4875 T^{4} + 96 p T^{5} + 98 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 356 T^{2} - 3472 T^{3} + 42678 T^{4} - 3472 p T^{5} + 356 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 414 T^{2} + 4872 T^{3} + 49254 T^{4} + 4872 p T^{5} + 414 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 200 T^{2} - 380 T^{3} + 20926 T^{4} - 380 p T^{5} + 200 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 382 T^{2} + 4376 T^{3} + 54838 T^{4} + 4376 p T^{5} + 382 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.22707724071875592262965732608, −7.14626714146478517031551690356, −6.90956902746447543731063996892, −6.80252045010260342613279818819, −6.72758381393065920972022996355, −6.16293742516790383516015289903, −5.89721466551016506161675691393, −5.41992874236598306750800661965, −5.25435526417244008306317293711, −5.01445195635644057834935280894, −4.61364346958868006280075538387, −4.58333595847102539294845652632, −4.42488196662679993420907122753, −4.01935192871305394257249077109, −3.85133179759958003787164771696, −3.81485383177270833811668517984, −3.69960074683715583883037142791, −3.42655711902715011767133599045, −3.19056387957771297649165857762, −2.68827018958856899147500052281, −2.47979099558138492163874274620, −2.15779425393411169085096375017, −1.68742286649753499374114587674, −1.59119139420269810191578961506, −1.28273354911826418250620422612, 0, 0, 0, 0, 1.28273354911826418250620422612, 1.59119139420269810191578961506, 1.68742286649753499374114587674, 2.15779425393411169085096375017, 2.47979099558138492163874274620, 2.68827018958856899147500052281, 3.19056387957771297649165857762, 3.42655711902715011767133599045, 3.69960074683715583883037142791, 3.81485383177270833811668517984, 3.85133179759958003787164771696, 4.01935192871305394257249077109, 4.42488196662679993420907122753, 4.58333595847102539294845652632, 4.61364346958868006280075538387, 5.01445195635644057834935280894, 5.25435526417244008306317293711, 5.41992874236598306750800661965, 5.89721466551016506161675691393, 6.16293742516790383516015289903, 6.72758381393065920972022996355, 6.80252045010260342613279818819, 6.90956902746447543731063996892, 7.14626714146478517031551690356, 7.22707724071875592262965732608

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