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Properties

Label 2-945-945.839-c1-0-30
Degree $2$
Conductor $945$
Sign $-0.720 - 0.693i$
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  + (−1.62 + 0.586i)3-s + (1.87 + 0.684i)4-s + (1.43 + 1.71i)5-s + (−2.48 + 0.904i)7-s + (2.31 − 1.91i)9-s + (−1.23 + 1.47i)11-s + (−3.46 − 0.0122i)12-s + (0.396 + 2.25i)13-s + (−3.34 − 1.94i)15-s + (3.06 + 2.57i)16-s + (−0.531 + 0.306i)17-s + (1.52 + 4.20i)20-s + (3.52 − 2.93i)21-s + (−0.868 + 4.92i)25-s + (−2.64 + 4.47i)27-s − 5.29·28-s + ⋯
L(s)  = 1  + (−0.940 + 0.338i)3-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)5-s + (−0.939 + 0.342i)7-s + (0.770 − 0.637i)9-s + (−0.373 + 0.445i)11-s + (−0.999 − 0.00354i)12-s + (0.110 + 0.624i)13-s + (−0.864 − 0.503i)15-s + (0.766 + 0.642i)16-s + (−0.128 + 0.0743i)17-s + (0.342 + 0.939i)20-s + (0.768 − 0.640i)21-s + (−0.173 + 0.984i)25-s + (−0.509 + 0.860i)27-s − 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.440507 + 1.09187i\)
\(L(\frac12)\) \(\approx\) \(0.440507 + 1.09187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 - 0.586i)T \)
5 \( 1 + (-1.43 - 1.71i)T \)
7 \( 1 + (2.48 - 0.904i)T \)
good2 \( 1 + (-1.87 - 0.684i)T^{2} \)
11 \( 1 + (1.23 - 1.47i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.396 - 2.25i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.531 - 0.306i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (4.76 + 0.839i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (1.18 + 3.25i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (13.1 - 7.61i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.67 - 4.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.06 - 17.3i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (16.9 + 2.98i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.0 - 12.6i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.28465403947136464989416880218, −9.902169898065081514681080640563, −8.910797500745055353161626848893, −7.39081417515491545044218181093, −6.83976698495108530427643672702, −6.13930663703909568579060822994, −5.48212210721219850425039874678, −4.01468800887101962359692194524, −2.97070077251509818136176083719, −1.88111593457634533684095563386, 0.58130192625261980115108935593, 1.81431033549513810267778035302, 3.13172058670332499483673421935, 4.67706841935319030780125533772, 5.79803081936592396207767547949, 6.00154300071884512507702548370, 7.03495838836099545504731411031, 7.79378830546018905401459476875, 9.050055383142346122557771855096, 10.07002060871706950474301369458

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