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Properties

Label 2-945-945.664-c0-0-0
Degree $2$
Conductor $945$
Sign $-0.957 + 0.286i$
Analytic cond. $0.471616$
Root an. cond. $0.686743$
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  + (−0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)7-s + (−0.499 − 0.866i)9-s + (−1.43 + 0.524i)11-s + (−0.939 − 0.342i)12-s + (−1.43 + 1.20i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)16-s + (−0.173 − 0.300i)17-s + (0.173 − 0.984i)20-s + (0.766 + 0.642i)21-s + (0.766 + 0.642i)25-s + 0.999·27-s + 0.999·28-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)7-s + (−0.499 − 0.866i)9-s + (−1.43 + 0.524i)11-s + (−0.939 − 0.342i)12-s + (−1.43 + 1.20i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)16-s + (−0.173 − 0.300i)17-s + (0.173 − 0.984i)20-s + (0.766 + 0.642i)21-s + (0.766 + 0.642i)25-s + 0.999·27-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.957 + 0.286i$
Analytic conductor: \(0.471616\)
Root analytic conductor: \(0.686743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (664, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :0),\ -0.957 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2253614070\)
\(L(\frac12)\) \(\approx\) \(0.2253614070\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
good2 \( 1 + (-0.173 - 0.984i)T^{2} \)
11 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
13 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
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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

−10.88924724088155005502013933445, −9.865964669445868582459896440220, −9.156669837084561821503665449899, −7.957860539945841406813317761203, −7.49993364330183428071269915341, −6.71104099090316780352404822133, −5.00977755790235461537305611892, −4.52380931055766701075916426465, −3.78200351071981139674530372738, −2.57114776049785082597813757767, 0.21099797944916853897416715208, 2.16730396020920085948346071451, 2.97651963860622448758819652527, 5.07115998228714712044108620063, 5.31118069735767719029496101270, 6.29950633754830136002906280604, 7.34798722489863090134931918664, 7.901780320473544278607882087767, 8.767265944996353980404653978447, 10.13954454863591534793197434629

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