↑

Properties

Label 2-315-315.103-c1-0-32
Degree $2$
Conductor $315$
Sign $0.930 + 0.366i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 0.366i)2-s + (1.5 − 0.866i)3-s + 1.73i·4-s + (1.23 − 1.86i)5-s + (0.232 − 0.866i)6-s + (−0.866 + 2.5i)7-s + (1.36 + 1.36i)8-s + (1.5 − 2.59i)9-s + (−0.232 − 1.13i)10-s + (1 − 1.73i)11-s + (1.49 + 2.59i)12-s + (0.0980 + 0.366i)13-s + (0.598 + 1.23i)14-s + (0.232 − 3.86i)15-s − 2.46·16-s + (−0.535 + 2i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.258i)2-s + (0.866 − 0.499i)3-s + 0.866i·4-s + (0.550 − 0.834i)5-s + (0.0947 − 0.353i)6-s + (−0.327 + 0.944i)7-s + (0.482 + 0.482i)8-s + (0.5 − 0.866i)9-s + (−0.0733 − 0.358i)10-s + (0.301 − 0.522i)11-s + (0.433 + 0.749i)12-s + (0.0272 + 0.101i)13-s + (0.159 + 0.329i)14-s + (0.0599 − 0.998i)15-s − 0.616·16-s + (−0.129 + 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.930 + 0.366i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.930 + 0.366i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02251 - 0.384374i\)
\(L(\frac12)\) \(\approx\) \(2.02251 - 0.384374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 + 1.86i)T \)
7 \( 1 + (0.866 - 2.5i)T \)
good2 \( 1 + (-0.366 + 0.366i)T - 2iT^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0980 - 0.366i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.535 - 2i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.366 - 0.633i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.63 + 6.09i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (8.59 - 4.96i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + (2.36 + 8.83i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.59 - 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.232 - 0.866i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.63 - 3.63i)T + 47iT^{2} \)
53 \( 1 + (-0.267 + i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + 9.12T + 59T^{2} \)
61 \( 1 - 10.9iT - 61T^{2} \)
67 \( 1 + (-2.46 + 2.46i)T - 67iT^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 + (12.9 + 3.46i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + (-9.96 - 2.66i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.535 + 0.928i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.43 - 9.09i)T + (-84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.11027697641016282452136136812, −10.78884846411091639617308992302, −9.192820830423544116819756501702, −8.887501930227689706472068401326, −8.061075703228973972945944961678, −6.81923272892507492237727041591, −5.64562113995676952097901305651, −4.19816802089695976760273790875, −3.00939325994860998949424756780, −1.88341844662801733029715740647, 1.89712901819148144537649828625, 3.43521235228360260303980207587, 4.53910515989732264836057147150, 5.79675996674074115427596972682, 6.97025666109637453231350398094, 7.59401869122062962830019246519, 9.403869665376179894242444817842, 9.713792170928113322224939075253, 10.57034288944233889008246227797, 11.35043249702888632017839214161

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