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Properties

Label 2-3736-3736.371-c0-0-0
Degree $2$
Conductor $3736$
Sign $-0.885 + 0.464i$
Analytic cond. $1.86450$
Root an. cond. $1.36546$
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  + (−0.0740 − 0.997i)2-s + (−0.627 − 1.55i)3-s + (−0.989 + 0.147i)4-s + (−1.50 + 0.740i)6-s + (0.220 + 0.975i)8-s + (−1.29 + 1.25i)9-s + (1.69 + 1.01i)11-s + (0.849 + 1.44i)12-s + (0.956 − 0.292i)16-s + (1.11 + 1.49i)17-s + (1.34 + 1.19i)18-s + (−0.718 − 1.58i)19-s + (0.885 − 1.76i)22-s + (1.37 − 0.954i)24-s + (−0.979 − 0.200i)25-s + ⋯
L(s)  = 1  + (−0.0740 − 0.997i)2-s + (−0.627 − 1.55i)3-s + (−0.989 + 0.147i)4-s + (−1.50 + 0.740i)6-s + (0.220 + 0.975i)8-s + (−1.29 + 1.25i)9-s + (1.69 + 1.01i)11-s + (0.849 + 1.44i)12-s + (0.956 − 0.292i)16-s + (1.11 + 1.49i)17-s + (1.34 + 1.19i)18-s + (−0.718 − 1.58i)19-s + (0.885 − 1.76i)22-s + (1.37 − 0.954i)24-s + (−0.979 − 0.200i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $-0.885 + 0.464i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ -0.885 + 0.464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9400554520\)
\(L(\frac12)\) \(\approx\) \(0.9400554520\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0740 + 0.997i)T \)
467 \( 1 + (0.755 - 0.655i)T \)
good3 \( 1 + (0.627 + 1.55i)T + (-0.718 + 0.695i)T^{2} \)
5 \( 1 + (0.979 + 0.200i)T^{2} \)
7 \( 1 + (0.999 + 0.0404i)T^{2} \)
11 \( 1 + (-1.69 - 1.01i)T + (0.472 + 0.881i)T^{2} \)
13 \( 1 + (0.259 - 0.965i)T^{2} \)
17 \( 1 + (-1.11 - 1.49i)T + (-0.285 + 0.958i)T^{2} \)
19 \( 1 + (0.718 + 1.58i)T + (-0.660 + 0.750i)T^{2} \)
23 \( 1 + (-0.990 - 0.134i)T^{2} \)
29 \( 1 + (0.984 - 0.174i)T^{2} \)
31 \( 1 + (0.913 + 0.405i)T^{2} \)
37 \( 1 + (0.362 + 0.932i)T^{2} \)
41 \( 1 + (-1.11 + 1.23i)T + (-0.100 - 0.994i)T^{2} \)
43 \( 1 + (-1.08 + 1.67i)T + (-0.412 - 0.911i)T^{2} \)
47 \( 1 + (0.680 - 0.732i)T^{2} \)
53 \( 1 + (0.0202 - 0.999i)T^{2} \)
59 \( 1 + (-1.23 - 0.413i)T + (0.797 + 0.602i)T^{2} \)
61 \( 1 + (0.960 - 0.279i)T^{2} \)
67 \( 1 + (-0.307 + 0.0291i)T + (0.982 - 0.187i)T^{2} \)
71 \( 1 + (-0.728 + 0.685i)T^{2} \)
73 \( 1 + (0.342 - 0.880i)T + (-0.737 - 0.675i)T^{2} \)
79 \( 1 + (0.575 + 0.817i)T^{2} \)
83 \( 1 + (-0.448 + 0.569i)T + (-0.233 - 0.972i)T^{2} \)
89 \( 1 + (-0.905 - 0.344i)T + (0.746 + 0.665i)T^{2} \)
97 \( 1 + (-1.16 + 1.61i)T + (-0.311 - 0.950i)T^{2} \)
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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

−8.455141003122945575209461621379, −7.55501719486935413794858890298, −6.98277502295393076163827410957, −6.17130694568070932046103351749, −5.52017900165672433565367303964, −4.36791543133195864622456307558, −3.71833343391306483609248950214, −2.28436611967686454672994795425, −1.76534128416338028060362448436, −0.815754129565005439202998169833, 1.03998756891827672967510115077, 3.30722419851274748944612719762, 3.85189210590003749608910817525, 4.51751309273045348753084786064, 5.35496370925484576603949997608, 6.06796506222082810350455336722, 6.36358909342471687519941235402, 7.63568812809500474393057278525, 8.276761146184655446530087102806, 9.331473708939052348123279897896

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