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Properties

Label 2-315-35.4-c1-0-17
Degree $2$
Conductor $315$
Sign $-0.0598 + 0.998i$
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  + (2.17 − 1.25i)2-s + (2.16 − 3.75i)4-s + (−2.23 − 0.00136i)5-s + (1.31 − 2.29i)7-s − 5.87i·8-s + (−4.87 + 2.81i)10-s + (0.489 − 0.847i)11-s + 5.14i·13-s + (−0.0275 − 6.65i)14-s + (−3.05 − 5.29i)16-s + (3.59 + 2.07i)17-s + (−1.15 − 1.99i)19-s + (−4.84 + 8.38i)20-s − 2.46i·22-s + (−4.39 + 2.53i)23-s + ⋯
L(s)  = 1  + (1.54 − 0.889i)2-s + (1.08 − 1.87i)4-s + (−0.999 − 0.000610i)5-s + (0.496 − 0.868i)7-s − 2.07i·8-s + (−1.54 + 0.888i)10-s + (0.147 − 0.255i)11-s + 1.42i·13-s + (−0.00735 − 1.77i)14-s + (−0.763 − 1.32i)16-s + (0.871 + 0.503i)17-s + (−0.263 − 0.456i)19-s + (−1.08 + 1.87i)20-s − 0.524i·22-s + (−0.916 + 0.528i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0598 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0598 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0598 + 0.998i$
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: \((2,\ 315,\ (\ :1/2),\ -0.0598 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81917 - 1.93145i\)
\(L(\frac12)\) \(\approx\) \(1.81917 - 1.93145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.00136i)T \)
7 \( 1 + (-1.31 + 2.29i)T \)
good2 \( 1 + (-2.17 + 1.25i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-0.489 + 0.847i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.14iT - 13T^{2} \)
17 \( 1 + (-3.59 - 2.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.15 + 1.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.39 - 2.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.92T + 29T^{2} \)
31 \( 1 + (0.316 - 0.548i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.84 - 4.52i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.65T + 41T^{2} \)
43 \( 1 - 0.344iT - 43T^{2} \)
47 \( 1 + (3.67 - 2.11i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.61 - 3.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.908 - 1.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.328 + 0.568i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.01 + 4.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.49T + 71T^{2} \)
73 \( 1 + (4.65 + 2.68i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.44 + 9.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.62iT - 83T^{2} \)
89 \( 1 + (8.15 + 14.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.53iT - 97T^{2} \)
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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

−11.82311660224079059822690739867, −10.86045474360272915495359264070, −10.11787074939907041591357506997, −8.543499519179871867959990570302, −7.29950702295714369839782415429, −6.29991872971431551456777724678, −4.85272374253337413923715801356, −4.15976696429291594100779656579, −3.31374110879371315223747446259, −1.55156253083244907405275545008, 2.82084025606871875378820219297, 3.87926093687224146738694748663, 5.01671240211457826874394971064, 5.71789186121212151558154648682, 6.92892368709050762056405786360, 7.929375276640026415964039284211, 8.434962002324622643518380245536, 10.25992449878156100466362684915, 11.52996529419111918076612439648, 12.31135812675929597756104568191

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