Properties

Label 2-315-7.3-c2-0-14
Degree $2$
Conductor $315$
Sign $-0.986 + 0.160i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.76 − 3.04i)2-s + (−4.19 + 7.26i)4-s + (1.93 − 1.11i)5-s + (0.244 + 6.99i)7-s + 15.4·8-s + (−6.81 − 3.93i)10-s + (1.29 − 2.24i)11-s − 11.5i·13-s + (20.8 − 13.0i)14-s + (−10.4 − 18.0i)16-s + (−20.0 − 11.6i)17-s + (25.9 − 14.9i)19-s + 18.7i·20-s − 9.13·22-s + (−17.5 − 30.4i)23-s + ⋯
L(s)  = 1  + (−0.880 − 1.52i)2-s + (−1.04 + 1.81i)4-s + (0.387 − 0.223i)5-s + (0.0348 + 0.999i)7-s + 1.93·8-s + (−0.681 − 0.393i)10-s + (0.117 − 0.204i)11-s − 0.890i·13-s + (1.49 − 0.932i)14-s + (−0.652 − 1.12i)16-s + (−1.18 − 0.682i)17-s + (1.36 − 0.788i)19-s + 0.938i·20-s − 0.415·22-s + (−0.763 − 1.32i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.986 + 0.160i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.986 + 0.160i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0640377 - 0.790746i\)
\(L(\frac12)\) \(\approx\) \(0.0640377 - 0.790746i\)
\(L(2)\) 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 \( 1 + (-1.93 + 1.11i)T \)
7 \( 1 + (-0.244 - 6.99i)T \)
good2 \( 1 + (1.76 + 3.04i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (-1.29 + 2.24i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 11.5iT - 169T^{2} \)
17 \( 1 + (20.0 + 11.6i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-25.9 + 14.9i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (17.5 + 30.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 24.4T + 841T^{2} \)
31 \( 1 + (32.4 + 18.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (12.8 + 22.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 3.71iT - 1.68e3T^{2} \)
43 \( 1 - 74.2T + 1.84e3T^{2} \)
47 \( 1 + (2.92 - 1.68i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (20.0 - 34.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-42.7 - 24.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (0.765 - 0.441i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-32.5 + 56.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 86.0T + 5.04e3T^{2} \)
73 \( 1 + (53.3 + 30.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (13.7 + 23.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 131. iT - 6.88e3T^{2} \)
89 \( 1 + (-56.5 + 32.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 42.2iT - 9.40e3T^{2} \)
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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

−10.99913214531092364986163042779, −10.10221399488983145472349529790, −9.124980020208181707493352940219, −8.769026124799091672626264402212, −7.56457569136438139614870459974, −5.92480538665788014842495128337, −4.61398680768214994549951007694, −3.01333675422428165578990140884, −2.20186481452888995731601463468, −0.53976035201160986702370566882, 1.45176051864671167295225043253, 3.96571254822751236280642356978, 5.29552624611802647416862450089, 6.37535961173381785044381537725, 7.11247628222619937651053690413, 7.86719537952701455455963121634, 8.982872574484194104637964974201, 9.753433998685507172162154506736, 10.50127613395885787435538874882, 11.65821610798888381363245758127

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