Properties

Label 2-2e3-8.3-c62-0-11
Degree $2$
Conductor $8$
Sign $1$
Analytic cond. $194.756$
Root an. cond. $13.9555$
Motivic weight $62$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.14e9·2-s − 1.10e15·3-s + 4.61e18·4-s + 2.37e24·6-s − 9.90e27·8-s + 8.43e29·9-s − 2.19e32·11-s − 5.10e33·12-s + 2.12e37·16-s − 2.69e38·17-s − 1.81e39·18-s + 1.86e39·19-s + 4.70e41·22-s + 1.09e43·24-s + 2.16e43·25-s − 5.10e44·27-s − 4.56e46·32-s + 2.42e47·33-s + 5.79e47·34-s + 3.88e48·36-s − 4.00e48·38-s − 7.48e49·41-s + 8.13e50·43-s − 1.01e51·44-s − 2.35e52·48-s + 2.48e52·49-s − 4.65e52·50-s + ⋯
L(s)  = 1  − 2-s − 1.79·3-s + 4-s + 1.79·6-s − 8-s + 2.20·9-s − 1.14·11-s − 1.79·12-s + 16-s − 1.93·17-s − 2.20·18-s + 0.425·19-s + 1.14·22-s + 1.79·24-s + 25-s − 2.16·27-s − 32-s + 2.04·33-s + 1.93·34-s + 2.20·36-s − 0.425·38-s − 0.754·41-s + 1.87·43-s − 1.14·44-s − 1.79·48-s + 49-s − 50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(63-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+31) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $1$
Analytic conductor: \(194.756\)
Root analytic conductor: \(13.9555\)
Motivic weight: \(62\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :31),\ 1)\)

Particular Values

\(L(\frac{63}{2})\) \(\approx\) \(0.1990784876\)
\(L(\frac12)\) \(\approx\) \(0.1990784876\)
\(L(32)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{31} T \)
good3 \( 1 + 1106589796438898 T + p^{62} T^{2} \)
5 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
7 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
11 \( 1 + \)\(21\!\cdots\!86\)\( T + p^{62} T^{2} \)
13 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
17 \( 1 + \)\(26\!\cdots\!02\)\( T + p^{62} T^{2} \)
19 \( 1 - \)\(18\!\cdots\!66\)\( T + p^{62} T^{2} \)
23 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
29 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
31 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
37 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
41 \( 1 + \)\(74\!\cdots\!46\)\( T + p^{62} T^{2} \)
43 \( 1 - \)\(81\!\cdots\!86\)\( T + p^{62} T^{2} \)
47 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
53 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
59 \( 1 + \)\(15\!\cdots\!82\)\( T + p^{62} T^{2} \)
61 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
67 \( 1 + \)\(58\!\cdots\!62\)\( T + p^{62} T^{2} \)
71 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
73 \( 1 + \)\(63\!\cdots\!58\)\( T + p^{62} T^{2} \)
79 \( ( 1 - p^{31} T )( 1 + p^{31} T ) \)
83 \( 1 + \)\(60\!\cdots\!58\)\( T + p^{62} T^{2} \)
89 \( 1 - \)\(53\!\cdots\!46\)\( T + p^{62} T^{2} \)
97 \( 1 - \)\(21\!\cdots\!94\)\( T + p^{62} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.91704419162737643313929990188, −10.48418080713287983232278238675, −9.048658263296277425106830628684, −7.49307030262866447039512765204, −6.58359078375601761605324348140, −5.62199213909310443551481879697, −4.53507258093847892189838992300, −2.58820295985976050036845333167, −1.32086912849988127243100050645, −0.25980509838338912446674407737, 0.25980509838338912446674407737, 1.32086912849988127243100050645, 2.58820295985976050036845333167, 4.53507258093847892189838992300, 5.62199213909310443551481879697, 6.58359078375601761605324348140, 7.49307030262866447039512765204, 9.048658263296277425106830628684, 10.48418080713287983232278238675, 10.91704419162737643313929990188

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