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Properties

Label 2-315-315.52-c1-0-12
Degree $2$
Conductor $315$
Sign $0.702 + 0.711i$
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  + (−1.36 − 1.36i)2-s + (1.5 − 0.866i)3-s + 1.73i·4-s + (−0.133 + 2.23i)5-s + (−3.23 − 0.866i)6-s + (2 + 1.73i)7-s + (−0.366 + 0.366i)8-s + (1.5 − 2.59i)9-s + (3.23 − 2.86i)10-s + (2.73 + 4.73i)11-s + (1.49 + 2.59i)12-s + (1 − 3.73i)13-s + (−0.366 − 5.09i)14-s + (1.73 + 3.46i)15-s + 4.46·16-s + (−0.732 − 2.73i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.965i)2-s + (0.866 − 0.499i)3-s + 0.866i·4-s + (−0.0599 + 0.998i)5-s + (−1.31 − 0.353i)6-s + (0.755 + 0.654i)7-s + (−0.129 + 0.129i)8-s + (0.5 − 0.866i)9-s + (1.02 − 0.906i)10-s + (0.823 + 1.42i)11-s + (0.433 + 0.749i)12-s + (0.277 − 1.03i)13-s + (−0.0978 − 1.36i)14-s + (0.447 + 0.894i)15-s + 1.11·16-s + (−0.177 − 0.662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06365 - 0.444375i\)
\(L(\frac12)\) \(\approx\) \(1.06365 - 0.444375i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (0.133 - 2.23i)T \)
7 \( 1 + (-2 - 1.73i)T \)
good2 \( 1 + (1.36 + 1.36i)T + 2iT^{2} \)
11 \( 1 + (-2.73 - 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 3.73i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.732 + 2.73i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.63 - 2.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.23 - 4.59i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (6 + 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.196iT - 31T^{2} \)
37 \( 1 + (-1.09 + 4.09i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.19 - 1.26i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.401 + 1.5i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-9.29 + 9.29i)T - 47iT^{2} \)
53 \( 1 + (1.63 + 6.09i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 + (-0.901 - 0.901i)T + 67iT^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (8.83 - 2.36i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + 4.53iT - 79T^{2} \)
83 \( 1 + (1.36 - 0.366i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-4.59 - 7.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.09 - 7.83i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.65156565204422681705779517961, −10.44634354054030879698209636522, −9.652733745412925308586662510163, −8.955645839324955095374697287038, −7.83013209222033954948963273177, −7.24215249684585579332716434953, −5.71244270739327673230280687041, −3.73052820374428214637474593329, −2.54387019424675151150326865726, −1.65257937697496838385957618899, 1.29028602696991915967275935970, 3.65635652176926386080733605700, 4.66775759763220961767394829433, 6.11969512133881678930208071563, 7.33384513408846383350110244655, 8.236482186205813697222391911248, 8.895630326548943972802933167358, 9.250991118974983309380382146040, 10.59820031666879158039119379015, 11.50774686754646020295724218996

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