Properties

Label 8-1078e4-1.1-c1e4-0-1
Degree $8$
Conductor $1.350\times 10^{12}$
Sign $1$
Analytic cond. $5490.14$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s + 2·8-s + 4·9-s + 2·11-s − 4·16-s − 8·18-s − 4·22-s − 12·23-s − 8·25-s + 8·29-s + 2·32-s + 4·36-s + 20·37-s − 32·43-s + 2·44-s + 24·46-s + 16·50-s − 16·53-s − 16·58-s + 3·64-s − 4·67-s − 8·71-s + 8·72-s − 40·74-s − 32·79-s + 9·81-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.707·8-s + 4/3·9-s + 0.603·11-s − 16-s − 1.88·18-s − 0.852·22-s − 2.50·23-s − 8/5·25-s + 1.48·29-s + 0.353·32-s + 2/3·36-s + 3.28·37-s − 4.87·43-s + 0.301·44-s + 3.53·46-s + 2.26·50-s − 2.19·53-s − 2.10·58-s + 3/8·64-s − 0.488·67-s − 0.949·71-s + 0.942·72-s − 4.64·74-s − 3.60·79-s + 81-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5490.14\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1078} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 7^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1530502242\)
\(L(\frac12)\) \(\approx\) \(0.1530502242\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7 \( 1 \)
11$C_2$ \( ( 1 - T + T^{2} )^{2} \)
good3$C_2^3$ \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
5$C_2^3$ \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2^3$ \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
31$C_2^3$ \( 1 - 60 T^{2} + 2639 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 - 76 T^{2} + 3567 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 116 T^{2} + 9975 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^3$ \( 1 - 114 T^{2} + 9275 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
73$C_2^3$ \( 1 - 74 T^{2} + 147 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + 96 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.14514558720728711668102465556, −6.96075729797655424594791704208, −6.51913572396882816143215733480, −6.47817717615397359755867292861, −6.35847628994209351256829494358, −6.13149707476088979412324054824, −5.66249895702367805770359502572, −5.58745505324211400281561294195, −5.46319089850240842925834156012, −4.74174954866515649927987989756, −4.70911807006801719758268150721, −4.47590798445976548610636401238, −4.34360222680365389620298447198, −4.08065735938503601543205085087, −3.96766298188808790951595826413, −3.30474698474347804759164360260, −3.17280420684963810221051575129, −3.16051524306858387648923877109, −2.41866996381734089683057062447, −2.20744775122042189013747546161, −1.70242775264785880087716475833, −1.63731177589728058753183449524, −1.39861236430928194746645600693, −0.863390962916596040672524380146, −0.12877449719525281497634525229, 0.12877449719525281497634525229, 0.863390962916596040672524380146, 1.39861236430928194746645600693, 1.63731177589728058753183449524, 1.70242775264785880087716475833, 2.20744775122042189013747546161, 2.41866996381734089683057062447, 3.16051524306858387648923877109, 3.17280420684963810221051575129, 3.30474698474347804759164360260, 3.96766298188808790951595826413, 4.08065735938503601543205085087, 4.34360222680365389620298447198, 4.47590798445976548610636401238, 4.70911807006801719758268150721, 4.74174954866515649927987989756, 5.46319089850240842925834156012, 5.58745505324211400281561294195, 5.66249895702367805770359502572, 6.13149707476088979412324054824, 6.35847628994209351256829494358, 6.47817717615397359755867292861, 6.51913572396882816143215733480, 6.96075729797655424594791704208, 7.14514558720728711668102465556

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