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Properties

Label 2-740-740.203-c0-0-0
Degree $2$
Conductor $740$
Sign $0.402 + 0.915i$
Analytic cond. $0.369308$
Root an. cond. $0.607707$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 − 0.342i)5-s + (0.500 − 0.866i)8-s + (−0.984 − 0.173i)9-s + (−0.939 − 0.342i)10-s + (−0.300 − 1.70i)13-s + (−0.939 + 0.342i)16-s + (1.85 + 0.326i)17-s + (0.642 + 0.766i)18-s + (0.499 + 0.866i)20-s + (0.766 − 0.642i)25-s + (−0.866 + 1.5i)26-s + (0.424 + 1.58i)29-s + (0.939 + 0.342i)32-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (0.939 − 0.342i)5-s + (0.500 − 0.866i)8-s + (−0.984 − 0.173i)9-s + (−0.939 − 0.342i)10-s + (−0.300 − 1.70i)13-s + (−0.939 + 0.342i)16-s + (1.85 + 0.326i)17-s + (0.642 + 0.766i)18-s + (0.499 + 0.866i)20-s + (0.766 − 0.642i)25-s + (−0.866 + 1.5i)26-s + (0.424 + 1.58i)29-s + (0.939 + 0.342i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7454049575\)
\(L(\frac12)\) \(\approx\) \(0.7454049575\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
good3 \( 1 + (0.984 + 0.173i)T^{2} \)
7 \( 1 + (-0.642 + 0.766i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.300 + 1.70i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-1.85 - 0.326i)T + (0.939 + 0.342i)T^{2} \)
19 \( 1 + (-0.984 - 0.173i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.424 - 1.58i)T + (-0.866 + 0.5i)T^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (0.673 - 0.118i)T + (0.939 - 0.342i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.469 + 0.218i)T + (0.642 + 0.766i)T^{2} \)
59 \( 1 + (-0.642 - 0.766i)T^{2} \)
61 \( 1 + (0.657 - 0.939i)T + (-0.342 - 0.939i)T^{2} \)
67 \( 1 + (0.642 - 0.766i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 + (0.642 - 0.766i)T^{2} \)
83 \( 1 + (-0.342 + 0.939i)T^{2} \)
89 \( 1 + (0.766 - 1.64i)T + (-0.642 - 0.766i)T^{2} \)
97 \( 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2} \)
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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

−10.29776744569413910367237493448, −9.746758031704562688262585902362, −8.725406834184771288192408357594, −8.179925549647441831291120289873, −7.17805794646791810437837496436, −5.83349196701273292691492765263, −5.22072874071745622412873736648, −3.43660816832380720117629417738, −2.69465447410502764290323074582, −1.16952663504902888006865938130, 1.68215166914304830762483112233, 2.87569636221968225739938126366, 4.71897853176888414933725195004, 5.72564433105895538204688455209, 6.33167879203969423404567351207, 7.27959131436953335663125500516, 8.184921411907399112112546221942, 9.142667816413862388465947885479, 9.721785065154202841368552316871, 10.42013829865048491058343455980

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