Properties

Label 2-1078-1.1-c1-0-12
Degree $2$
Conductor $1078$
Sign $1$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.82·3-s + 4-s − 2.82·6-s − 8-s + 5.00·9-s − 11-s + 2.82·12-s − 2.82·13-s + 16-s + 2.82·17-s − 5.00·18-s + 5.65·19-s + 22-s + 8·23-s − 2.82·24-s − 5·25-s + 2.82·26-s + 5.65·27-s + 2·29-s + 8.48·31-s − 32-s − 2.82·33-s − 2.82·34-s + 5.00·36-s + 2·37-s − 5.65·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.63·3-s + 0.5·4-s − 1.15·6-s − 0.353·8-s + 1.66·9-s − 0.301·11-s + 0.816·12-s − 0.784·13-s + 0.250·16-s + 0.685·17-s − 1.17·18-s + 1.29·19-s + 0.213·22-s + 1.66·23-s − 0.577·24-s − 25-s + 0.554·26-s + 1.08·27-s + 0.371·29-s + 1.52·31-s − 0.176·32-s − 0.492·33-s − 0.485·34-s + 0.833·36-s + 0.328·37-s − 0.917·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.124644924\)
\(L(\frac12)\) \(\approx\) \(2.124644924\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 2.82T + 3T^{2} \)
5 \( 1 + 5T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 + 8.48T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + 16.9T + 97T^{2} \)
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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

−9.820477154514011232455265934263, −8.997466491481587343659289507775, −8.326665340597870716505581679439, −7.51717193146548534533156801639, −7.12777322153850334744972367874, −5.66182867008759703605509091147, −4.44963573958408879388959489379, −3.12631326230837995645522018306, −2.65856383754343291430511836413, −1.28019551589000328596134817038, 1.28019551589000328596134817038, 2.65856383754343291430511836413, 3.12631326230837995645522018306, 4.44963573958408879388959489379, 5.66182867008759703605509091147, 7.12777322153850334744972367874, 7.51717193146548534533156801639, 8.326665340597870716505581679439, 8.997466491481587343659289507775, 9.820477154514011232455265934263

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