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Properties

Label 2-740-740.647-c0-0-0
Degree $2$
Conductor $740$
Sign $0.125 + 0.992i$
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  + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.500i)8-s + (0.984 + 0.173i)9-s + (0.173 − 0.984i)10-s + (−1.70 + 0.300i)13-s + (−0.939 + 0.342i)16-s + (−0.326 + 1.85i)17-s + (0.766 − 0.642i)18-s + (−0.642 − 0.766i)20-s + (0.499 − 0.866i)25-s + (−0.866 + 1.5i)26-s + (−0.424 − 1.58i)29-s + (−0.342 + 0.939i)32-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.500i)8-s + (0.984 + 0.173i)9-s + (0.173 − 0.984i)10-s + (−1.70 + 0.300i)13-s + (−0.939 + 0.342i)16-s + (−0.326 + 1.85i)17-s + (0.766 − 0.642i)18-s + (−0.642 − 0.766i)20-s + (0.499 − 0.866i)25-s + (−0.866 + 1.5i)26-s + (−0.424 − 1.58i)29-s + (−0.342 + 0.939i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.125 + 0.992i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.125 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.423039164\)
\(L(\frac12)\) \(\approx\) \(1.423039164\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (-0.984 + 0.173i)T \)
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 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
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 \( 1 + (0.218 - 0.469i)T + (-0.642 - 0.766i)T^{2} \)
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 \( 1 + (0.984 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.15755499883754341522782530179, −9.909563819073490251793884775541, −9.039381325710201485052923063753, −7.79769784088941909813597759412, −6.57836800328517676616291001657, −5.78132140937798493924872256611, −4.71442471416957109258152600405, −4.12339197838747666165830876393, −2.46081872740519679359040361226, −1.62788211661275319159173968068, 2.30757495317234191123677241546, 3.31002131445263096759138357195, 4.83945806147790776002182000487, 5.19976800812298395071029964862, 6.60966417968564719509201788715, 7.05769068506949582909613047181, 7.80702442601289350997575941765, 9.332066479640458607171784861061, 9.605616683642995869435481682783, 10.73758359464878914998109867487

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