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Properties

Label 2-315-35.2-c2-0-20
Degree $2$
Conductor $315$
Sign $0.499 + 0.866i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.445 − 0.119i)2-s + (−3.28 + 1.89i)4-s + (−4.59 + 1.97i)5-s + (−0.171 − 6.99i)7-s + (−2.53 + 2.53i)8-s + (−1.81 + 1.42i)10-s + (7.70 + 13.3i)11-s + (17.1 − 17.1i)13-s + (−0.911 − 3.09i)14-s + (6.74 − 11.6i)16-s + (5.59 − 20.8i)17-s + (−15.6 − 9.01i)19-s + (11.3 − 15.1i)20-s + (5.02 + 5.02i)22-s + (−2.98 − 11.1i)23-s + ⋯
L(s)  = 1  + (0.222 − 0.0596i)2-s + (−0.820 + 0.473i)4-s + (−0.919 + 0.394i)5-s + (−0.0245 − 0.999i)7-s + (−0.317 + 0.317i)8-s + (−0.181 + 0.142i)10-s + (0.700 + 1.21i)11-s + (1.31 − 1.31i)13-s + (−0.0650 − 0.221i)14-s + (0.421 − 0.730i)16-s + (0.329 − 1.22i)17-s + (−0.822 − 0.474i)19-s + (0.567 − 0.758i)20-s + (0.228 + 0.228i)22-s + (−0.129 − 0.484i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.499 + 0.866i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.499 + 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.954056 - 0.551277i\)
\(L(\frac12)\) \(\approx\) \(0.954056 - 0.551277i\)
\(L(2)\) 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 \( 1 + (4.59 - 1.97i)T \)
7 \( 1 + (0.171 + 6.99i)T \)
good2 \( 1 + (-0.445 + 0.119i)T + (3.46 - 2i)T^{2} \)
11 \( 1 + (-7.70 - 13.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-17.1 + 17.1i)T - 169iT^{2} \)
17 \( 1 + (-5.59 + 20.8i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (15.6 + 9.01i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (2.98 + 11.1i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 1.87iT - 841T^{2} \)
31 \( 1 + (-9.52 - 16.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (4.95 - 1.32i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 51.4T + 1.68e3T^{2} \)
43 \( 1 + (-18.4 + 18.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (22.4 - 6.01i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (75.2 + 20.1i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-40.1 + 23.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-12.3 + 21.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-29.9 + 111. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 63.9T + 5.04e3T^{2} \)
73 \( 1 + (-13.3 - 3.56i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-45.3 - 26.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-34.3 + 34.3i)T - 6.88e3iT^{2} \)
89 \( 1 + (44.5 + 25.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-39.0 - 39.0i)T + 9.40e3iT^{2} \)
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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

−11.29256360229040843496042009901, −10.46735708798585159960133473086, −9.398315452027557348504845373879, −8.276845169453869268037300429997, −7.55091361513428947249646693947, −6.57490316375636333336359152237, −4.85892022809173170337342536384, −4.04604295030831524611111071888, −3.14322108763333008079515101895, −0.59113630713136125603991917366, 1.31841423746527822122240078899, 3.64872684642409299397295732614, 4.27416455911956171706425066820, 5.79414901605552474386433692373, 6.30839366287805564775172155170, 8.248748495791731453295698520656, 8.675059958477327855746068374494, 9.421863922241573310639607615085, 10.90471788891406015699302473451, 11.59458873889577949272340531739

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