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Properties

Label 2-737-737.428-c0-0-0
Degree $2$
Conductor $737$
Sign $0.791 - 0.610i$
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  + (0.698 − 0.449i)3-s + (−0.327 + 0.945i)4-s + (−0.0671 + 0.466i)5-s + (−0.128 + 0.281i)9-s + (0.928 − 0.371i)11-s + (0.195 + 0.807i)12-s + (0.162 + 0.356i)15-s + (−0.786 − 0.618i)16-s + (−0.419 − 0.216i)20-s + (0.0800 + 1.68i)23-s + (0.746 + 0.219i)25-s + (0.154 + 1.07i)27-s + (−1.44 − 1.37i)31-s + (0.481 − 0.676i)33-s + (−0.224 − 0.213i)36-s + (0.995 − 1.72i)37-s + ⋯
L(s)  = 1  + (0.698 − 0.449i)3-s + (−0.327 + 0.945i)4-s + (−0.0671 + 0.466i)5-s + (−0.128 + 0.281i)9-s + (0.928 − 0.371i)11-s + (0.195 + 0.807i)12-s + (0.162 + 0.356i)15-s + (−0.786 − 0.618i)16-s + (−0.419 − 0.216i)20-s + (0.0800 + 1.68i)23-s + (0.746 + 0.219i)25-s + (0.154 + 1.07i)27-s + (−1.44 − 1.37i)31-s + (0.481 − 0.676i)33-s + (−0.224 − 0.213i)36-s + (0.995 − 1.72i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(737\)    =    \(11 \cdot 67\)
Sign: $0.791 - 0.610i$
Analytic conductor: \(0.367810\)
Root analytic conductor: \(0.606474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{737} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 737,\ (\ :0),\ 0.791 - 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104012928\)
\(L(\frac12)\) \(\approx\) \(1.104012928\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.928 + 0.371i)T \)
67 \( 1 + (-0.0475 + 0.998i)T \)
good2 \( 1 + (0.327 - 0.945i)T^{2} \)
3 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (0.0671 - 0.466i)T + (-0.959 - 0.281i)T^{2} \)
7 \( 1 + (-0.981 + 0.189i)T^{2} \)
13 \( 1 + (-0.0475 + 0.998i)T^{2} \)
17 \( 1 + (0.786 - 0.618i)T^{2} \)
19 \( 1 + (-0.981 - 0.189i)T^{2} \)
23 \( 1 + (-0.0800 - 1.68i)T + (-0.995 + 0.0950i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (1.44 + 1.37i)T + (0.0475 + 0.998i)T^{2} \)
37 \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.928 - 0.371i)T^{2} \)
43 \( 1 + (0.142 + 0.989i)T^{2} \)
47 \( 1 + (1.03 + 0.531i)T + (0.580 + 0.814i)T^{2} \)
53 \( 1 + (1.21 + 1.40i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (1.78 - 0.523i)T + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.723 - 0.690i)T^{2} \)
71 \( 1 + (-0.627 + 1.81i)T + (-0.786 - 0.618i)T^{2} \)
73 \( 1 + (-0.723 - 0.690i)T^{2} \)
79 \( 1 + (0.888 - 0.458i)T^{2} \)
83 \( 1 + (-0.235 - 0.971i)T^{2} \)
89 \( 1 + (-1.21 - 0.782i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.94240721901377598460209478276, −9.305678165190459131731702493718, −9.111985698752948383194411134120, −7.82799040675485021696753813582, −7.58176849016936131653710195967, −6.52444691934398873468052792625, −5.25879312267623912077361587684, −3.85832952552740051483148059136, −3.22550291944334719803006158815, −2.00652595547490685114943758013, 1.33170745976490825633934786403, 2.91846693779702836747532282856, 4.24336416347045316240908137627, 4.81750192332226972561948868430, 6.11015432904802779838977593867, 6.82924761196264325281003220375, 8.319186652592969716930707521024, 8.943891652192974964800553101810, 9.488709651686390930841244774097, 10.30454620339993541772955146806

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