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Properties

Label 2-315-7.4-c1-0-12
Degree $2$
Conductor $315$
Sign $-0.863 + 0.504i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (−0.605 − 1.04i)2-s + (0.267 − 0.462i)4-s + (−0.5 − 0.866i)5-s + (1.16 − 2.37i)7-s − 3.06·8-s + (−0.605 + 1.04i)10-s + (−0.127 + 0.220i)11-s − 0.744·13-s + (−3.19 + 0.220i)14-s + (1.32 + 2.29i)16-s + (0.605 − 1.04i)17-s + (−0.556 − 0.963i)19-s − 0.534·20-s + 0.308·22-s + (−3.92 − 6.80i)23-s + ⋯
L(s)  = 1  + (−0.428 − 0.741i)2-s + (0.133 − 0.231i)4-s + (−0.223 − 0.387i)5-s + (0.439 − 0.898i)7-s − 1.08·8-s + (−0.191 + 0.331i)10-s + (−0.0384 + 0.0666i)11-s − 0.206·13-s + (−0.854 + 0.0590i)14-s + (0.330 + 0.573i)16-s + (0.146 − 0.254i)17-s + (−0.127 − 0.221i)19-s − 0.119·20-s + 0.0658·22-s + (−0.819 − 1.41i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.863 + 0.504i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.863 + 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245734 - 0.906812i\)
\(L(\frac12)\) \(\approx\) \(0.245734 - 0.906812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.16 + 2.37i)T \)
good2 \( 1 + (0.605 + 1.04i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (0.127 - 0.220i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.744T + 13T^{2} \)
17 \( 1 + (-0.605 + 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.556 + 0.963i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.92 + 6.80i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 + (3.45 - 5.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.63 + 8.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.81T + 41T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 + (-3.95 - 6.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.21 - 2.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.94 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.47 - 9.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.58 + 7.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (-0.217 + 0.377i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.18 + 2.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (-7.08 - 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.44T + 97T^{2} \)
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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

−10.96926980642346688376271166943, −10.58559411505742133351599316405, −9.551453528460367182979834608103, −8.636601373866546156709789047428, −7.55343401913117186209055176967, −6.43587088712437422605390908726, −5.10161020298328020267736153464, −3.90230584787921138751954715846, −2.31602201323315731397039842419, −0.77927690805501548934203885259, 2.36589161960345231968486167039, 3.73015919186361027607666652306, 5.43398599367824574218198695293, 6.27907928150395419792322100811, 7.43901661738830307728297896538, 8.107124332135372702388211215504, 8.994356235866489498310512925244, 9.978988820932015978433432521084, 11.33238857967455201435316824686, 11.89792844196060868847379953923

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