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Properties

Label 2-945-315.52-c1-0-30
Degree $2$
Conductor $945$
Sign $0.967 - 0.252i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + (1.36 + 1.36i)2-s + 1.73i·4-s + (0.133 − 2.23i)5-s + (2 + 1.73i)7-s + (0.366 − 0.366i)8-s + (3.23 − 2.86i)10-s + (−2.73 − 4.73i)11-s + (1 − 3.73i)13-s + (0.366 + 5.09i)14-s + 4.46·16-s + (0.732 + 2.73i)17-s + (1.63 + 2.83i)19-s + (3.86 + 0.232i)20-s + (2.73 − 10.1i)22-s + (−1.23 − 4.59i)23-s + ⋯
L(s)  = 1  + (0.965 + 0.965i)2-s + 0.866i·4-s + (0.0599 − 0.998i)5-s + (0.755 + 0.654i)7-s + (0.129 − 0.129i)8-s + (1.02 − 0.906i)10-s + (−0.823 − 1.42i)11-s + (0.277 − 1.03i)13-s + (0.0978 + 1.36i)14-s + 1.11·16-s + (0.177 + 0.662i)17-s + (0.374 + 0.649i)19-s + (0.864 + 0.0518i)20-s + (0.582 − 2.17i)22-s + (−0.256 − 0.958i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.967 - 0.252i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86292 + 0.368054i\)
\(L(\frac12)\) \(\approx\) \(2.86292 + 0.368054i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.133 + 2.23i)T \)
7 \( 1 + (-2 - 1.73i)T \)
good2 \( 1 + (-1.36 - 1.36i)T + 2iT^{2} \)
11 \( 1 + (2.73 + 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 3.73i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.732 - 2.73i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.63 - 2.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.23 + 4.59i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-6 - 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.196iT - 31T^{2} \)
37 \( 1 + (-1.09 + 4.09i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.19 + 1.26i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.401 + 1.5i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (9.29 - 9.29i)T - 47iT^{2} \)
53 \( 1 + (-1.63 - 6.09i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 + (-0.901 - 0.901i)T + 67iT^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (8.83 - 2.36i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + 4.53iT - 79T^{2} \)
83 \( 1 + (-1.36 + 0.366i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (4.59 + 7.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.09 - 7.83i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.19202316775477356422576608769, −8.789084229058319151621000492551, −8.201181173457492323388664679216, −7.74227803453644943088962726869, −6.17737593563805222677207327324, −5.66997441326423765285616827611, −5.11327642896754610685167979684, −4.14615366270349283635618205278, −2.94119650671258773915520040307, −1.12489855210463995951794313362, 1.72582892374349782045826031627, 2.56578502160998268475076252162, 3.65377983187856175180870843606, 4.59516363301560264398380788136, 5.18190048713282078997972095948, 6.64831776689306148015111330339, 7.37966218627612831081654833225, 8.131524848866043492858355972466, 9.801518702036283524627341778488, 10.08453447667599989685660254868

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