Properties

Label 8-3822e4-1.1-c1e4-0-3
Degree $8$
Conductor $2.134\times 10^{14}$
Sign $1$
Analytic cond. $867503.$
Root an. cond. $5.52438$
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  + 4·3-s − 2·4-s + 10·9-s − 8·12-s − 6·13-s + 3·16-s + 2·17-s − 2·23-s + 7·25-s + 20·27-s − 2·29-s − 20·36-s − 24·39-s − 34·43-s + 12·48-s + 8·51-s + 12·52-s + 36·53-s − 2·61-s − 4·64-s − 4·68-s − 8·69-s + 28·75-s + 64·79-s + 35·81-s − 8·87-s + 4·92-s + ⋯
L(s)  = 1  + 2.30·3-s − 4-s + 10/3·9-s − 2.30·12-s − 1.66·13-s + 3/4·16-s + 0.485·17-s − 0.417·23-s + 7/5·25-s + 3.84·27-s − 0.371·29-s − 3.33·36-s − 3.84·39-s − 5.18·43-s + 1.73·48-s + 1.12·51-s + 1.66·52-s + 4.94·53-s − 0.256·61-s − 1/2·64-s − 0.485·68-s − 0.963·69-s + 3.23·75-s + 7.20·79-s + 35/9·81-s − 0.857·87-s + 0.417·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(867503.\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.13103175\)
\(L(\frac12)\) \(\approx\) \(10.13103175\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
13$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
good5$C_2^3$ \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - p T^{2} + 64 T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 67 T^{2} + 1840 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + T + 8 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 20 T^{2} + 934 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 31 T^{2} - 120 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 17 T + 154 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 44 T^{2} + 3814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 68 T^{2} + 3766 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 + T + 118 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 216 T^{2} + 20030 T^{4} - 216 p^{2} T^{6} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 + 40 T^{2} + 4974 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 103 T^{2} + 11776 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
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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

−5.98791276603164938453349139854, −5.52851085819616010763122175276, −5.47973408174171856314477625130, −5.40487244296739256636806767877, −5.18591537604593243419644402671, −4.85071078499621171806624781608, −4.74954793759031444716002927456, −4.63692989768248469166880973328, −4.46435379890749963783489388658, −3.97950747940066417667162600432, −3.93205851924136161117812890696, −3.72293461044671651501968539682, −3.44456631299192908983119225726, −3.41691938851331204841228803633, −3.00112053517902904382370006538, −2.98036190595648597365047428641, −2.79491089641858014169331483239, −2.11524485355977788165632034388, −2.07473122963751526216263115121, −2.04978226664106756026810346045, −2.02052540619842198919323873327, −1.19603627635854026925673851246, −1.13747369718734960942181820643, −0.55749121441925118733232411534, −0.47482386793054703418209224041, 0.47482386793054703418209224041, 0.55749121441925118733232411534, 1.13747369718734960942181820643, 1.19603627635854026925673851246, 2.02052540619842198919323873327, 2.04978226664106756026810346045, 2.07473122963751526216263115121, 2.11524485355977788165632034388, 2.79491089641858014169331483239, 2.98036190595648597365047428641, 3.00112053517902904382370006538, 3.41691938851331204841228803633, 3.44456631299192908983119225726, 3.72293461044671651501968539682, 3.93205851924136161117812890696, 3.97950747940066417667162600432, 4.46435379890749963783489388658, 4.63692989768248469166880973328, 4.74954793759031444716002927456, 4.85071078499621171806624781608, 5.18591537604593243419644402671, 5.40487244296739256636806767877, 5.47973408174171856314477625130, 5.52851085819616010763122175276, 5.98791276603164938453349139854

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