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Properties

Label 2-3104-776.259-c0-0-0
Degree $2$
Conductor $3104$
Sign $-0.523 - 0.851i$
Analytic cond. $1.54909$
Root an. cond. $1.24462$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 + 0.158i)3-s + (0.465 − 0.124i)9-s + (1.17 + 1.53i)11-s + (−1.57 − 0.534i)17-s + (−0.382 + 0.0761i)19-s + (−0.130 + 0.991i)25-s + (0.582 − 0.241i)27-s + (−1.66 − 1.66i)33-s + (1.95 + 0.128i)41-s + (−1.91 − 0.513i)43-s + (−0.793 + 0.608i)49-s + (1.98 + 0.394i)51-s + (0.449 − 0.152i)57-s + (−0.996 − 0.491i)59-s + (0.389 + 1.95i)67-s + ⋯
L(s)  = 1  + (−1.20 + 0.158i)3-s + (0.465 − 0.124i)9-s + (1.17 + 1.53i)11-s + (−1.57 − 0.534i)17-s + (−0.382 + 0.0761i)19-s + (−0.130 + 0.991i)25-s + (0.582 − 0.241i)27-s + (−1.66 − 1.66i)33-s + (1.95 + 0.128i)41-s + (−1.91 − 0.513i)43-s + (−0.793 + 0.608i)49-s + (1.98 + 0.394i)51-s + (0.449 − 0.152i)57-s + (−0.996 − 0.491i)59-s + (0.389 + 1.95i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3104\)    =    \(2^{5} \cdot 97\)
Sign: $-0.523 - 0.851i$
Analytic conductor: \(1.54909\)
Root analytic conductor: \(1.24462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3104} (1423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3104,\ (\ :0),\ -0.523 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5200357266\)
\(L(\frac12)\) \(\approx\) \(0.5200357266\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 + (0.991 - 0.130i)T \)
good3 \( 1 + (1.20 - 0.158i)T + (0.965 - 0.258i)T^{2} \)
5 \( 1 + (0.130 - 0.991i)T^{2} \)
7 \( 1 + (0.793 - 0.608i)T^{2} \)
11 \( 1 + (-1.17 - 1.53i)T + (-0.258 + 0.965i)T^{2} \)
13 \( 1 + (-0.130 + 0.991i)T^{2} \)
17 \( 1 + (1.57 + 0.534i)T + (0.793 + 0.608i)T^{2} \)
19 \( 1 + (0.382 - 0.0761i)T + (0.923 - 0.382i)T^{2} \)
23 \( 1 + (-0.608 + 0.793i)T^{2} \)
29 \( 1 + (0.991 + 0.130i)T^{2} \)
31 \( 1 + (0.965 - 0.258i)T^{2} \)
37 \( 1 + (0.608 + 0.793i)T^{2} \)
41 \( 1 + (-1.95 - 0.128i)T + (0.991 + 0.130i)T^{2} \)
43 \( 1 + (1.91 + 0.513i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.258 + 0.965i)T^{2} \)
59 \( 1 + (0.996 + 0.491i)T + (0.608 + 0.793i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.389 - 1.95i)T + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (-0.991 + 0.130i)T^{2} \)
73 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (0.284 - 0.837i)T + (-0.793 - 0.608i)T^{2} \)
89 \( 1 + (0.0999 - 0.241i)T + (-0.707 - 0.707i)T^{2} \)
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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

−9.307464555120229540639932222822, −8.488671042010364369580826397183, −7.31217236455379207208531378313, −6.77705909990225858950282082965, −6.21618131754265106212567244006, −5.23439682333590426331139296009, −4.56539998902024891229202916127, −3.97742856967178051401590773994, −2.49571177421627777904925971508, −1.42022198341106382219542678069, 0.38693582084575045672180767396, 1.67775310415337207426577502824, 3.03722089307497564143859685834, 4.08076183343830602073085637365, 4.78519407278151713021963601608, 5.83631533123921610778230811832, 6.40462594845237490423644904559, 6.62637121368448101958307015006, 7.943402427858433962542001439686, 8.701805397837258835727263812305

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