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Properties

Label 2-740-740.383-c0-0-0
Degree $2$
Conductor $740$
Sign $0.764 - 0.644i$
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  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.866 + 0.5i)5-s + (0.866 + 0.5i)8-s + (0.342 + 0.939i)9-s + (−0.939 + 0.342i)10-s + (0.592 − 1.62i)13-s + (0.766 + 0.642i)16-s + (−1.43 + 0.524i)17-s + (0.173 + 0.984i)18-s + (−0.984 + 0.173i)20-s + (0.499 − 0.866i)25-s + (0.866 − 1.5i)26-s + (−0.816 + 0.218i)29-s + (0.642 + 0.766i)32-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.866 + 0.5i)5-s + (0.866 + 0.5i)8-s + (0.342 + 0.939i)9-s + (−0.939 + 0.342i)10-s + (0.592 − 1.62i)13-s + (0.766 + 0.642i)16-s + (−1.43 + 0.524i)17-s + (0.173 + 0.984i)18-s + (−0.984 + 0.173i)20-s + (0.499 − 0.866i)25-s + (0.866 − 1.5i)26-s + (−0.816 + 0.218i)29-s + (0.642 + 0.766i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.542607754\)
\(L(\frac12)\) \(\approx\) \(1.542607754\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 - 0.173i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (-0.342 + 0.939i)T \)
good3 \( 1 + (-0.342 - 0.939i)T^{2} \)
7 \( 1 + (0.984 + 0.173i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
19 \( 1 + (-0.342 - 0.939i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.816 - 0.218i)T + (0.866 - 0.5i)T^{2} \)
31 \( 1 + iT^{2} \)
41 \( 1 + (-0.439 + 1.20i)T + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.168 + 1.92i)T + (-0.984 + 0.173i)T^{2} \)
59 \( 1 + (-0.984 + 0.173i)T^{2} \)
61 \( 1 + (1.64 - 0.766i)T + (0.642 - 0.766i)T^{2} \)
67 \( 1 + (-0.984 - 0.173i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
79 \( 1 + (0.984 + 0.173i)T^{2} \)
83 \( 1 + (-0.642 - 0.766i)T^{2} \)
89 \( 1 + (-0.173 - 1.98i)T + (-0.984 + 0.173i)T^{2} \)
97 \( 1 + (0.342 + 0.592i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.91514695417948754677744863679, −10.30122719400437131432966773205, −8.606775238626609185708893790634, −7.84746669332765959047675377519, −7.20810738661568882024042344905, −6.19265706003014955362726252774, −5.22087214487291229796573507897, −4.22046012538914399754228271971, −3.36184895739625839583905019400, −2.19702367912220988542991785592, 1.57231095733345363035232617912, 3.16738692510039186285834946474, 4.29578448807597526264068318682, 4.57096043062824988502342279596, 6.15333947361185046594758580438, 6.76219825226390380544567842302, 7.66942698320789075568262022680, 8.940042802451927332511392618042, 9.511640512286199479088442674560, 10.93685831820392977520878477882

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