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Properties

Label 2-945-945.524-c1-0-137
Degree $2$
Conductor $945$
Sign $-0.734 - 0.678i$
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  + (−0.306 − 1.70i)3-s + (−0.347 − 1.96i)4-s + (0.764 − 2.10i)5-s + (−0.459 + 2.60i)7-s + (−2.81 + 1.04i)9-s + (−0.890 − 2.44i)11-s + (−3.25 + 1.19i)12-s + (−0.146 + 0.123i)13-s + (−3.81 − 0.659i)15-s + (−3.75 + 1.36i)16-s + (−7.10 + 4.10i)17-s + (−4.40 − 0.776i)20-s + (4.58 − 0.0162i)21-s + (−3.83 − 3.21i)25-s + (2.64 + 4.47i)27-s + 5.29·28-s + ⋯
L(s)  = 1  + (−0.177 − 0.984i)3-s + (−0.173 − 0.984i)4-s + (0.342 − 0.939i)5-s + (−0.173 + 0.984i)7-s + (−0.937 + 0.348i)9-s + (−0.268 − 0.737i)11-s + (−0.938 + 0.345i)12-s + (−0.0407 + 0.0342i)13-s + (−0.985 − 0.170i)15-s + (−0.939 + 0.342i)16-s + (−1.72 + 0.994i)17-s + (−0.984 − 0.173i)20-s + (0.999 − 0.00354i)21-s + (−0.766 − 0.642i)25-s + (0.509 + 0.860i)27-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.734 - 0.678i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219524 + 0.561248i\)
\(L(\frac12)\) \(\approx\) \(0.219524 + 0.561248i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.306 + 1.70i)T \)
5 \( 1 + (-0.764 + 2.10i)T \)
7 \( 1 + (0.459 - 2.60i)T \)
good2 \( 1 + (0.347 + 1.96i)T^{2} \)
11 \( 1 + (0.890 + 2.44i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.146 - 0.123i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (7.10 - 4.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-6.91 + 8.23i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (7.67 + 1.35i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-8.45 + 4.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.52 - 13.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.32 + 7.82i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-10.9 + 13.0i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.18 + 1.15i)T + (74.3 - 62.3i)T^{2} \)
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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

−9.352113421071108844416894279184, −8.614120099836168823865264325787, −8.190424764157636452723677718711, −6.54008060216021570104573707543, −6.12295906633084811677839024053, −5.38509990790086239736885935740, −4.46521316158216547741683099316, −2.52277610080633501958130048453, −1.66620747219316351074579460454, −0.27286909897360051148981462881, 2.53439138099397553297484462724, 3.35321850295305517358447427485, 4.34027365925709025897179040955, 5.00146748277296709675109976320, 6.64095563641148852313684123051, 6.97867393880050386455446411977, 8.063534560422900426838743107044, 9.096247588521154806984575244065, 9.754612957998905477535639687879, 10.62147408699669789846355276340

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