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Properties

Label 2-740-740.463-c0-0-0
Degree $2$
Conductor $740$
Sign $-0.991 + 0.129i$
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 + (−1.10 + 0.296i)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 + (−1.10 + 0.296i)29-s + (0.342 − 0.939i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.991 + 0.129i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ -0.991 + 0.129i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2160406191\)
\(L(\frac12)\) \(\approx\) \(0.2160406191\)
\(L(1)\) 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 \( 1 + (1.10 - 0.296i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (-1.75 - 0.816i)T + (0.642 + 0.766i)T^{2} \)
59 \( 1 + (0.642 + 0.766i)T^{2} \)
61 \( 1 + (1.34 + 0.939i)T + (0.342 + 0.939i)T^{2} \)
67 \( 1 + (0.642 - 0.766i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
79 \( 1 + (-0.642 + 0.766i)T^{2} \)
83 \( 1 + (-0.342 + 0.939i)T^{2} \)
89 \( 1 + (-0.766 - 0.357i)T + (0.642 + 0.766i)T^{2} \)
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

−10.78573242845632052298450444410, −10.19565576859877230127883895140, −9.077945622766835542890790410631, −8.396855831102107880701440872705, −7.57617842656709222780475678907, −6.85555154275597116354539113067, −5.92168592939420545764735099763, −4.89405850244638940842002075876, −3.71093115109255188414317923255, −2.20053531237598777928723014866, 0.26425341263058568824718342544, 2.35959192368561298468016594127, 3.27384209442318755550405282664, 4.57362919981585083632500902903, 5.32334068625903488577332558593, 7.23126773648736526316478578562, 7.54049558681938444147841112023, 8.657586401099017395626706294932, 9.211803544473543828639454450513, 10.10419928121821147805369176242

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