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Properties

Label 2-737-737.659-c0-0-0
Degree $2$
Conductor $737$
Sign $0.539 + 0.842i$
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  + (−0.544 − 1.19i)3-s + (−0.786 + 0.618i)4-s + (1.70 − 0.500i)5-s + (−0.468 + 0.540i)9-s + (0.723 + 0.690i)11-s + (1.16 + 0.600i)12-s + (−1.52 − 1.75i)15-s + (0.235 − 0.971i)16-s + (−1.03 + 1.44i)20-s + (−0.827 − 0.0789i)23-s + (1.81 − 1.16i)25-s + (−0.357 − 0.105i)27-s + (0.0934 − 1.96i)31-s + (0.428 − 1.23i)33-s + (0.0340 − 0.714i)36-s + (−0.981 + 1.70i)37-s + ⋯
L(s)  = 1  + (−0.544 − 1.19i)3-s + (−0.786 + 0.618i)4-s + (1.70 − 0.500i)5-s + (−0.468 + 0.540i)9-s + (0.723 + 0.690i)11-s + (1.16 + 0.600i)12-s + (−1.52 − 1.75i)15-s + (0.235 − 0.971i)16-s + (−1.03 + 1.44i)20-s + (−0.827 − 0.0789i)23-s + (1.81 − 1.16i)25-s + (−0.357 − 0.105i)27-s + (0.0934 − 1.96i)31-s + (0.428 − 1.23i)33-s + (0.0340 − 0.714i)36-s + (−0.981 + 1.70i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(737\)    =    \(11 \cdot 67\)
Sign: $0.539 + 0.842i$
Analytic conductor: \(0.367810\)
Root analytic conductor: \(0.606474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{737} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 737,\ (\ :0),\ 0.539 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9171272489\)
\(L(\frac12)\) \(\approx\) \(0.9171272489\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.723 - 0.690i)T \)
67 \( 1 + (0.995 - 0.0950i)T \)
good2 \( 1 + (0.786 - 0.618i)T^{2} \)
3 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
5 \( 1 + (-1.70 + 0.500i)T + (0.841 - 0.540i)T^{2} \)
7 \( 1 + (-0.928 - 0.371i)T^{2} \)
13 \( 1 + (0.995 - 0.0950i)T^{2} \)
17 \( 1 + (-0.235 - 0.971i)T^{2} \)
19 \( 1 + (-0.928 + 0.371i)T^{2} \)
23 \( 1 + (0.827 + 0.0789i)T + (0.981 + 0.189i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.0934 + 1.96i)T + (-0.995 - 0.0950i)T^{2} \)
37 \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.723 + 0.690i)T^{2} \)
43 \( 1 + (0.959 + 0.281i)T^{2} \)
47 \( 1 + (0.379 - 0.532i)T + (-0.327 - 0.945i)T^{2} \)
53 \( 1 + (0.205 + 1.43i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (-1.21 - 0.782i)T + (0.415 + 0.909i)T^{2} \)
61 \( 1 + (-0.0475 + 0.998i)T^{2} \)
71 \( 1 + (1.32 - 1.04i)T + (0.235 - 0.971i)T^{2} \)
73 \( 1 + (-0.0475 + 0.998i)T^{2} \)
79 \( 1 + (-0.580 - 0.814i)T^{2} \)
83 \( 1 + (0.888 + 0.458i)T^{2} \)
89 \( 1 + (-0.0395 + 0.0865i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.12560307020202547916279451130, −9.608332954359715806204524920048, −8.782841983271353332886354935319, −7.83346163762398787671469099952, −6.81129530462622517800085971659, −6.08248642388105074323681277869, −5.24846163367370329122572963855, −4.16012390555400088364957210821, −2.37274733608426455280472503810, −1.31585374269621423602568668769, 1.71239564710703301069093858980, 3.42017331239007849364704779200, 4.52375712870992498006915131155, 5.55773691508191345848555155435, 5.82254185887037339780630479875, 6.87463398295522283921155130556, 8.748566710137750219443082354712, 9.211602811394323260146380512115, 10.03823346740367578886835686616, 10.44186042719292165883156118328

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