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Properties

Label 2-3716-3716.319-c0-0-0
Degree $2$
Conductor $3716$
Sign $0.882 - 0.470i$
Analytic cond. $1.85452$
Root an. cond. $1.36180$
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  + (−0.970 + 0.241i)2-s + (0.883 − 0.468i)4-s + (−1.62 + 0.0879i)5-s + (−0.744 + 0.667i)8-s + (−0.241 + 0.970i)9-s + (1.55 − 0.476i)10-s + (−0.0103 − 0.765i)13-s + (0.561 − 0.827i)16-s + (−1.28 − 1.06i)17-s − 0.999i·18-s + (−1.39 + 0.837i)20-s + (1.62 − 0.177i)25-s + (0.194 + 0.740i)26-s + (0.0745 + 0.202i)29-s + (−0.344 + 0.938i)32-s + ⋯
L(s)  = 1  + (−0.970 + 0.241i)2-s + (0.883 − 0.468i)4-s + (−1.62 + 0.0879i)5-s + (−0.744 + 0.667i)8-s + (−0.241 + 0.970i)9-s + (1.55 − 0.476i)10-s + (−0.0103 − 0.765i)13-s + (0.561 − 0.827i)16-s + (−1.28 − 1.06i)17-s − 0.999i·18-s + (−1.39 + 0.837i)20-s + (1.62 − 0.177i)25-s + (0.194 + 0.740i)26-s + (0.0745 + 0.202i)29-s + (−0.344 + 0.938i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3716\)    =    \(2^{2} \cdot 929\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(1.85452\)
Root analytic conductor: \(1.36180\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3716} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3716,\ (\ :0),\ 0.882 - 0.470i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4585069125\)
\(L(\frac12)\) \(\approx\) \(0.4585069125\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.970 - 0.241i)T \)
929 \( 1 + iT \)
good3 \( 1 + (0.241 - 0.970i)T^{2} \)
5 \( 1 + (1.62 - 0.0879i)T + (0.994 - 0.108i)T^{2} \)
7 \( 1 + (-0.538 + 0.842i)T^{2} \)
11 \( 1 + (-0.998 - 0.0541i)T^{2} \)
13 \( 1 + (0.0103 + 0.765i)T + (-0.999 + 0.0270i)T^{2} \)
17 \( 1 + (1.28 + 1.06i)T + (0.188 + 0.982i)T^{2} \)
19 \( 1 + (-0.0541 - 0.998i)T^{2} \)
23 \( 1 + (-0.468 + 0.883i)T^{2} \)
29 \( 1 + (-0.0745 - 0.202i)T + (-0.762 + 0.647i)T^{2} \)
31 \( 1 + (-0.344 + 0.938i)T^{2} \)
37 \( 1 + (0.150 - 0.0600i)T + (0.725 - 0.687i)T^{2} \)
41 \( 1 + (-1.92 + 0.397i)T + (0.918 - 0.395i)T^{2} \)
43 \( 1 + (0.241 - 0.970i)T^{2} \)
47 \( 1 + (-0.419 - 0.907i)T^{2} \)
53 \( 1 + (-0.208 - 1.91i)T + (-0.976 + 0.214i)T^{2} \)
59 \( 1 + (0.870 + 0.492i)T^{2} \)
61 \( 1 + (-0.0456 - 1.68i)T + (-0.998 + 0.0541i)T^{2} \)
67 \( 1 + (-0.395 + 0.918i)T^{2} \)
71 \( 1 + (0.214 + 0.976i)T^{2} \)
73 \( 1 + (0.554 + 0.950i)T + (-0.492 + 0.870i)T^{2} \)
79 \( 1 + (-0.779 - 0.626i)T^{2} \)
83 \( 1 + (0.419 + 0.907i)T^{2} \)
89 \( 1 + (-0.497 + 0.198i)T + (0.725 - 0.687i)T^{2} \)
97 \( 1 + (-1.81 - 0.172i)T + (0.982 + 0.188i)T^{2} \)
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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

−8.812796141841902066558409585698, −7.77794218502588272082724263293, −7.61740649853826766214559313679, −6.93249403711867540141929456403, −5.90592795627933285333715656481, −4.98996039849991587413055494208, −4.21809290810191232662471056141, −3.03279010283574162830859037888, −2.33552981927211396236290682098, −0.68940529939468610472292184499, 0.60946376935703453508092896181, 2.00154544650327544883936043276, 3.16991269794402507036414226020, 3.92408521463236181367596686546, 4.45322974569121580654717193012, 6.02142656654200165579914355965, 6.67351699974817380440670840563, 7.29951985489414258487496288625, 8.084653106134588577908894423980, 8.637663334228499540787043601890

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