Properties

Label 2-315-35.4-c1-0-14
Degree $2$
Conductor $315$
Sign $-0.341 + 0.939i$
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  + (−0.248 + 0.143i)2-s + (−0.958 + 1.66i)4-s + (−0.717 − 2.11i)5-s + (−1.11 − 2.39i)7-s − 1.12i·8-s + (0.482 + 0.423i)10-s + (−1.66 + 2.88i)11-s − 4.54i·13-s + (0.622 + 0.436i)14-s + (−1.75 − 3.04i)16-s + (−4.80 − 2.77i)17-s + (−0.828 − 1.43i)19-s + (4.20 + 0.838i)20-s − 0.956i·22-s + (6.61 − 3.81i)23-s + ⋯
L(s)  = 1  + (−0.175 + 0.101i)2-s + (−0.479 + 0.830i)4-s + (−0.321 − 0.947i)5-s + (−0.421 − 0.906i)7-s − 0.397i·8-s + (0.152 + 0.134i)10-s + (−0.502 + 0.869i)11-s − 1.26i·13-s + (0.166 + 0.116i)14-s + (−0.438 − 0.760i)16-s + (−1.16 − 0.672i)17-s + (−0.190 − 0.329i)19-s + (0.940 + 0.187i)20-s − 0.204i·22-s + (1.37 − 0.795i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.341 + 0.939i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.341 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.329682 - 0.470698i\)
\(L(\frac12)\) \(\approx\) \(0.329682 - 0.470698i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.717 + 2.11i)T \)
7 \( 1 + (1.11 + 2.39i)T \)
good2 \( 1 + (0.248 - 0.143i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (1.66 - 2.88i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.54iT - 13T^{2} \)
17 \( 1 + (4.80 + 2.77i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.828 + 1.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.61 + 3.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.118T + 29T^{2} \)
31 \( 1 + (-3.13 + 5.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.71 - 3.87i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.0701T + 41T^{2} \)
43 \( 1 - 2.92iT - 43T^{2} \)
47 \( 1 + (5.53 - 3.19i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.640 - 0.369i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.815 + 1.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.65 - 6.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.62 - 1.51i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.77T + 71T^{2} \)
73 \( 1 + (2.03 + 1.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.97 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.22iT - 83T^{2} \)
89 \( 1 + (-6.50 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.04iT - 97T^{2} \)
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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

−11.45490153796688247600914237855, −10.30693197341814231411396474592, −9.381785909576725150691008468897, −8.484344838457947466441064290104, −7.64342790283116247114910874113, −6.82670110129190520028201498827, −4.97675617534828114165397120739, −4.33828753910644243437313484550, −2.96492247710435361293317699250, −0.43145365452909247784102442715, 2.12213697125758185722406287825, 3.55750777727916903456767530788, 5.03530890989494049049142827116, 6.15147914412564382998758565571, 6.88393008047770380863205858274, 8.506417018821899975629671641793, 9.061588999327619983309512414548, 10.16515270192651780324306194445, 11.00566897213209225960956572906, 11.61207731352590704006272062469

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