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Properties

Label 2-740-185.17-c1-0-9
Degree $2$
Conductor $740$
Sign $0.0332 - 0.999i$
Analytic cond. $5.90892$
Root an. cond. $2.43082$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.37 + 1.66i)3-s + (2.22 − 0.176i)5-s + (0.449 + 5.14i)7-s + (1.84 − 5.06i)9-s + (3.78 − 2.18i)11-s + (6.33 − 2.30i)13-s + (−4.99 + 4.12i)15-s + (−0.665 + 1.82i)17-s + (−0.587 − 0.838i)19-s + (−9.60 − 11.4i)21-s + (−0.711 + 1.23i)23-s + (4.93 − 0.786i)25-s + (1.78 + 6.67i)27-s + (−0.983 + 3.67i)29-s + (2.70 − 2.70i)31-s + ⋯
L(s)  = 1  + (−1.36 + 0.958i)3-s + (0.996 − 0.0788i)5-s + (0.170 + 1.94i)7-s + (0.613 − 1.68i)9-s + (1.14 − 0.658i)11-s + (1.75 − 0.639i)13-s + (−1.28 + 1.06i)15-s + (−0.161 + 0.443i)17-s + (−0.134 − 0.192i)19-s + (−2.09 − 2.49i)21-s + (−0.148 + 0.256i)23-s + (0.987 − 0.157i)25-s + (0.343 + 1.28i)27-s + (−0.182 + 0.681i)29-s + (0.485 − 0.485i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0332 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0332 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.0332 - 0.999i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ 0.0332 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.957874 + 0.926550i\)
\(L(\frac12)\) \(\approx\) \(0.957874 + 0.926550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.22 + 0.176i)T \)
37 \( 1 + (-4.62 + 3.95i)T \)
good3 \( 1 + (2.37 - 1.66i)T + (1.02 - 2.81i)T^{2} \)
7 \( 1 + (-0.449 - 5.14i)T + (-6.89 + 1.21i)T^{2} \)
11 \( 1 + (-3.78 + 2.18i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.33 + 2.30i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.665 - 1.82i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.587 + 0.838i)T + (-6.49 + 17.8i)T^{2} \)
23 \( 1 + (0.711 - 1.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.983 - 3.67i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.70 + 2.70i)T - 31iT^{2} \)
41 \( 1 + (-1.10 - 3.04i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 - 1.12T + 43T^{2} \)
47 \( 1 + (6.86 - 1.83i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.734 - 8.39i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (0.537 - 6.14i)T + (-58.1 - 10.2i)T^{2} \)
61 \( 1 + (1.34 - 2.88i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (5.21 - 0.456i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (-1.89 + 10.7i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (8.08 + 8.08i)T + 73iT^{2} \)
79 \( 1 + (-8.71 + 0.762i)T + (77.7 - 13.7i)T^{2} \)
83 \( 1 + (7.42 - 3.46i)T + (53.3 - 63.5i)T^{2} \)
89 \( 1 + (10.5 + 0.920i)T + (87.6 + 15.4i)T^{2} \)
97 \( 1 + (5.56 + 3.21i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.83263412545829307825481597056, −9.672188695835767492372154844783, −9.049455899973915868974318391766, −8.458219708376681748189311924677, −6.30580850550148148264761084031, −6.00095933488720663633923825418, −5.55900791133807540854240594921, −4.41643267638231838465177774277, −3.11395604917117952296571590870, −1.41951218568412235093482819713, 1.01962321011379377661494717673, 1.64846147020423656318421290592, 3.89368389243785980593607486212, 4.77899653753624046067897281506, 6.07910040951725838181128994165, 6.65186712177204086198701217897, 7.05348531054942231485000700501, 8.259971579842365402123770306227, 9.596601921181023065384631348551, 10.34498736041118249542443327449

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