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Properties

Label 2-3716-3716.327-c0-0-0
Degree $2$
Conductor $3716$
Sign $-0.231 - 0.972i$
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  + (0.870 + 0.492i)2-s + (0.515 + 0.856i)4-s + (1.37 + 1.44i)5-s + (0.0270 + 0.999i)8-s + (−0.492 − 0.870i)9-s + (0.481 + 1.93i)10-s + (0.154 − 0.749i)13-s + (−0.468 + 0.883i)16-s + (−0.916 + 1.23i)17-s − i·18-s + (−0.533 + 1.92i)20-s + (−0.160 + 2.96i)25-s + (0.503 − 0.576i)26-s + (−1.68 − 1.07i)29-s + (−0.842 + 0.538i)32-s + ⋯
L(s)  = 1  + (0.870 + 0.492i)2-s + (0.515 + 0.856i)4-s + (1.37 + 1.44i)5-s + (0.0270 + 0.999i)8-s + (−0.492 − 0.870i)9-s + (0.481 + 1.93i)10-s + (0.154 − 0.749i)13-s + (−0.468 + 0.883i)16-s + (−0.916 + 1.23i)17-s − i·18-s + (−0.533 + 1.92i)20-s + (−0.160 + 2.96i)25-s + (0.503 − 0.576i)26-s + (−1.68 − 1.07i)29-s + (−0.842 + 0.538i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3716\)    =    \(2^{2} \cdot 929\)
Sign: $-0.231 - 0.972i$
Analytic conductor: \(1.85452\)
Root analytic conductor: \(1.36180\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3716} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3716,\ (\ :0),\ -0.231 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.579738086\)
\(L(\frac12)\) \(\approx\) \(2.579738086\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.870 - 0.492i)T \)
929 \( 1 + iT \)
good3 \( 1 + (0.492 + 0.870i)T^{2} \)
5 \( 1 + (-1.37 - 1.44i)T + (-0.0541 + 0.998i)T^{2} \)
7 \( 1 + (0.779 - 0.626i)T^{2} \)
11 \( 1 + (-0.687 + 0.725i)T^{2} \)
13 \( 1 + (-0.154 + 0.749i)T + (-0.918 - 0.395i)T^{2} \)
17 \( 1 + (0.916 - 1.23i)T + (-0.293 - 0.955i)T^{2} \)
19 \( 1 + (0.725 - 0.687i)T^{2} \)
23 \( 1 + (0.856 + 0.515i)T^{2} \)
29 \( 1 + (1.68 + 1.07i)T + (0.419 + 0.907i)T^{2} \)
31 \( 1 + (-0.842 + 0.538i)T^{2} \)
37 \( 1 + (-1.55 + 1.05i)T + (0.370 - 0.928i)T^{2} \)
41 \( 1 + (-1.74 + 0.166i)T + (0.982 - 0.188i)T^{2} \)
43 \( 1 + (0.492 + 0.870i)T^{2} \)
47 \( 1 + (0.214 - 0.976i)T^{2} \)
53 \( 1 + (-1.20 - 0.0655i)T + (0.994 + 0.108i)T^{2} \)
59 \( 1 + (0.135 - 0.990i)T^{2} \)
61 \( 1 + (-0.495 + 1.15i)T + (-0.687 - 0.725i)T^{2} \)
67 \( 1 + (-0.188 + 0.982i)T^{2} \)
71 \( 1 + (0.108 - 0.994i)T^{2} \)
73 \( 1 + (1.27 + 0.0861i)T + (0.990 + 0.135i)T^{2} \)
79 \( 1 + (0.744 - 0.667i)T^{2} \)
83 \( 1 + (-0.214 + 0.976i)T^{2} \)
89 \( 1 + (-1.31 + 0.893i)T + (0.370 - 0.928i)T^{2} \)
97 \( 1 + (-0.296 + 1.97i)T + (-0.955 - 0.293i)T^{2} \)
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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

−8.963655878648578112092556006101, −7.88105071345513523349410309163, −7.23037743821425360973029109214, −6.38125681206848660562263342530, −5.90431392969490277337520612048, −5.68418918820815923427015616194, −4.20523212983189231386142841105, −3.47532596916902966164612547184, −2.64717261854598395719623588819, −1.99219921345663555891713203478, 1.13826949905962452901603906269, 2.10664209532665092483185111505, 2.63154789140371984203911430714, 4.13563214717539372892509941561, 4.79845526157115885417364304240, 5.32007756586648689491854425939, 5.93714638486056176407128728755, 6.71809891152458824023512949508, 7.72232105091247771332308977134, 8.870329602834962319496952062471

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