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Properties

Label 2-737-737.505-c0-0-0
Degree $2$
Conductor $737$
Sign $0.685 - 0.727i$
Analytic cond. $0.367810$
Root an. cond. $0.606474$
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  + (0.186 + 0.215i)3-s + (0.235 + 0.971i)4-s + (0.975 + 0.627i)5-s + (0.130 − 0.909i)9-s + (0.0475 − 0.998i)11-s + (−0.165 + 0.231i)12-s + (0.0469 + 0.326i)15-s + (−0.888 + 0.458i)16-s + (−0.379 + 1.09i)20-s + (−1.28 + 0.247i)23-s + (0.143 + 0.314i)25-s + (0.459 − 0.295i)27-s + (−1.84 + 0.176i)31-s + (0.223 − 0.175i)33-s + (0.914 − 0.0873i)36-s + (−0.928 + 1.60i)37-s + ⋯
L(s)  = 1  + (0.186 + 0.215i)3-s + (0.235 + 0.971i)4-s + (0.975 + 0.627i)5-s + (0.130 − 0.909i)9-s + (0.0475 − 0.998i)11-s + (−0.165 + 0.231i)12-s + (0.0469 + 0.326i)15-s + (−0.888 + 0.458i)16-s + (−0.379 + 1.09i)20-s + (−1.28 + 0.247i)23-s + (0.143 + 0.314i)25-s + (0.459 − 0.295i)27-s + (−1.84 + 0.176i)31-s + (0.223 − 0.175i)33-s + (0.914 − 0.0873i)36-s + (−0.928 + 1.60i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(737\)    =    \(11 \cdot 67\)
Sign: $0.685 - 0.727i$
Analytic conductor: \(0.367810\)
Root analytic conductor: \(0.606474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{737} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 737,\ (\ :0),\ 0.685 - 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.203570197\)
\(L(\frac12)\) \(\approx\) \(1.203570197\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.0475 + 0.998i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
good2 \( 1 + (-0.235 - 0.971i)T^{2} \)
3 \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \)
5 \( 1 + (-0.975 - 0.627i)T + (0.415 + 0.909i)T^{2} \)
7 \( 1 + (-0.723 + 0.690i)T^{2} \)
13 \( 1 + (-0.981 - 0.189i)T^{2} \)
17 \( 1 + (0.888 + 0.458i)T^{2} \)
19 \( 1 + (-0.723 - 0.690i)T^{2} \)
23 \( 1 + (1.28 - 0.247i)T + (0.928 - 0.371i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (1.84 - 0.176i)T + (0.981 - 0.189i)T^{2} \)
37 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.0475 - 0.998i)T^{2} \)
43 \( 1 + (-0.841 + 0.540i)T^{2} \)
47 \( 1 + (-0.514 + 1.48i)T + (-0.786 - 0.618i)T^{2} \)
53 \( 1 + (0.0913 + 0.0268i)T + (0.841 + 0.540i)T^{2} \)
59 \( 1 + (-0.0395 + 0.0865i)T + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (0.995 - 0.0950i)T^{2} \)
71 \( 1 + (-0.195 - 0.807i)T + (-0.888 + 0.458i)T^{2} \)
73 \( 1 + (0.995 - 0.0950i)T^{2} \)
79 \( 1 + (0.327 + 0.945i)T^{2} \)
83 \( 1 + (-0.580 + 0.814i)T^{2} \)
89 \( 1 + (-1.30 + 1.50i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.61063951069633223432186804827, −9.853004605655362505436153334730, −8.941557004625655755823500525740, −8.263726565309603859234931611598, −7.08593831437502522363817124712, −6.40799778827552023265108610262, −5.51603035912293382856631715416, −3.88329758946343036022581399916, −3.25573598265086945580105840983, −2.07923674563557950027010530554, 1.69720469005374246181726391599, 2.22626740501381137516715091970, 4.27788227559940457226270633776, 5.28840314255165518417415439860, 5.82375969363946503090559176607, 6.97734730540744393946584351633, 7.79279175122818504324860977392, 9.087465502340108692702424523315, 9.567161445403322738447131564639, 10.43761814282542202025858072047

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