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Properties

Label 2-315-21.20-c9-0-42
Degree $2$
Conductor $315$
Sign $0.102 + 0.994i$
Analytic cond. $162.236$
Root an. cond. $12.7372$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 38.5i·2-s − 971.·4-s + 625·5-s + (5.53e3 + 3.11e3i)7-s + 1.77e4i·8-s − 2.40e4i·10-s + 3.58e4i·11-s − 1.02e5i·13-s + (1.20e5 − 2.13e5i)14-s + 1.84e5·16-s + 2.29e5·17-s − 4.84e5i·19-s − 6.07e5·20-s + 1.38e6·22-s + 6.86e5i·23-s + ⋯
L(s)  = 1  − 1.70i·2-s − 1.89·4-s + 0.447·5-s + (0.871 + 0.490i)7-s + 1.52i·8-s − 0.761i·10-s + 0.738i·11-s − 1.00i·13-s + (0.835 − 1.48i)14-s + 0.705·16-s + 0.667·17-s − 0.852i·19-s − 0.848·20-s + 1.25·22-s + 0.511i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.102 + 0.994i$
Analytic conductor: \(162.236\)
Root analytic conductor: \(12.7372\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :9/2),\ 0.102 + 0.994i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.371286997\)
\(L(\frac12)\) \(\approx\) \(2.371286997\)
\(L(\frac{11}{2})\) 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 \( 1 - 625T \)
7 \( 1 + (-5.53e3 - 3.11e3i)T \)
good2 \( 1 + 38.5iT - 512T^{2} \)
11 \( 1 - 3.58e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.02e5iT - 1.06e10T^{2} \)
17 \( 1 - 2.29e5T + 1.18e11T^{2} \)
19 \( 1 + 4.84e5iT - 3.22e11T^{2} \)
23 \( 1 - 6.86e5iT - 1.80e12T^{2} \)
29 \( 1 - 2.83e6iT - 1.45e13T^{2} \)
31 \( 1 - 2.96e6iT - 2.64e13T^{2} \)
37 \( 1 + 1.64e7T + 1.29e14T^{2} \)
41 \( 1 + 1.37e7T + 3.27e14T^{2} \)
43 \( 1 - 1.57e7T + 5.02e14T^{2} \)
47 \( 1 - 2.20e7T + 1.11e15T^{2} \)
53 \( 1 + 5.73e7iT - 3.29e15T^{2} \)
59 \( 1 - 2.98e7T + 8.66e15T^{2} \)
61 \( 1 - 2.12e8iT - 1.16e16T^{2} \)
67 \( 1 - 1.05e8T + 2.72e16T^{2} \)
71 \( 1 + 1.24e8iT - 4.58e16T^{2} \)
73 \( 1 - 2.55e8iT - 5.88e16T^{2} \)
79 \( 1 - 2.67e8T + 1.19e17T^{2} \)
83 \( 1 + 1.62e8T + 1.86e17T^{2} \)
89 \( 1 - 7.43e8T + 3.50e17T^{2} \)
97 \( 1 - 4.95e8iT - 7.60e17T^{2} \)
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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

−10.18885112059590775438427041695, −9.218756210812206099690016817622, −8.439685957182477746545269714005, −7.17664785418072721357667174647, −5.44594025459696092739526175524, −4.83722748177342493068835648676, −3.52887783974627961781875822687, −2.55410709127012544809458994702, −1.70266195000373315928197998298, −0.835573627598076713625921363873, 0.57883155196224265992619547019, 1.89304964127468880341454734600, 3.77863896646317023136021454396, 4.76980448304055216812849689468, 5.67933827082066483169635069544, 6.47160611563778908000823095921, 7.44994197870128616483144567906, 8.212604192504959318699445740468, 8.985249635416960198321912131714, 10.03849842605192585883297545078

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