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Properties

Label 2-737-737.395-c0-0-0
Degree $2$
Conductor $737$
Sign $-0.128 + 0.991i$
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  + (−1.61 − 0.474i)3-s + (0.580 − 0.814i)4-s + (1.02 − 1.18i)5-s + (1.54 + 0.989i)9-s + (0.981 + 0.189i)11-s + (−1.32 + 1.04i)12-s + (−2.22 + 1.43i)15-s + (−0.327 − 0.945i)16-s + (−0.370 − 1.52i)20-s + (−1.38 + 1.32i)23-s + (−0.209 − 1.45i)25-s + (−0.915 − 1.05i)27-s + (0.0883 − 0.0353i)31-s + (−1.49 − 0.770i)33-s + (1.69 − 0.680i)36-s + (−0.0475 + 0.0824i)37-s + ⋯
L(s)  = 1  + (−1.61 − 0.474i)3-s + (0.580 − 0.814i)4-s + (1.02 − 1.18i)5-s + (1.54 + 0.989i)9-s + (0.981 + 0.189i)11-s + (−1.32 + 1.04i)12-s + (−2.22 + 1.43i)15-s + (−0.327 − 0.945i)16-s + (−0.370 − 1.52i)20-s + (−1.38 + 1.32i)23-s + (−0.209 − 1.45i)25-s + (−0.915 − 1.05i)27-s + (0.0883 − 0.0353i)31-s + (−1.49 − 0.770i)33-s + (1.69 − 0.680i)36-s + (−0.0475 + 0.0824i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(737\)    =    \(11 \cdot 67\)
Sign: $-0.128 + 0.991i$
Analytic conductor: \(0.367810\)
Root analytic conductor: \(0.606474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{737} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 737,\ (\ :0),\ -0.128 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8008533720\)
\(L(\frac12)\) \(\approx\) \(0.8008533720\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.981 - 0.189i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
good2 \( 1 + (-0.580 + 0.814i)T^{2} \)
3 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
5 \( 1 + (-1.02 + 1.18i)T + (-0.142 - 0.989i)T^{2} \)
7 \( 1 + (0.995 + 0.0950i)T^{2} \)
13 \( 1 + (-0.723 - 0.690i)T^{2} \)
17 \( 1 + (0.327 - 0.945i)T^{2} \)
19 \( 1 + (0.995 - 0.0950i)T^{2} \)
23 \( 1 + (1.38 - 1.32i)T + (0.0475 - 0.998i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.0883 + 0.0353i)T + (0.723 - 0.690i)T^{2} \)
37 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.981 + 0.189i)T^{2} \)
43 \( 1 + (0.654 + 0.755i)T^{2} \)
47 \( 1 + (0.419 + 1.72i)T + (-0.888 + 0.458i)T^{2} \)
53 \( 1 + (-0.815 - 1.78i)T + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.279 - 1.94i)T + (-0.959 - 0.281i)T^{2} \)
61 \( 1 + (-0.928 + 0.371i)T^{2} \)
71 \( 1 + (0.165 - 0.231i)T + (-0.327 - 0.945i)T^{2} \)
73 \( 1 + (-0.928 + 0.371i)T^{2} \)
79 \( 1 + (-0.235 + 0.971i)T^{2} \)
83 \( 1 + (0.786 - 0.618i)T^{2} \)
89 \( 1 + (1.78 - 0.523i)T + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.25873924680092293651870235553, −9.828780591562127248244080836807, −8.869141922739698072741206297593, −7.39600889036163621493033204476, −6.49757282836905719801030237423, −5.79187048832671317567384776198, −5.39764604125237782025669101971, −4.36589451638110875259662066316, −1.86711090748898397290393723698, −1.17090733415714998701305698336, 2.02161964333161422215167784346, 3.39692006634437321955421053555, 4.48470401339691196931970084248, 5.80444855563037832396026798729, 6.52525097046579932945025100364, 6.72140653042142452875334802403, 8.105627494973520976821171005957, 9.506163784701470030879951683148, 10.20234919786067734909311493572, 10.94183338648845311856454618275

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