↑

Properties

Label 2-117-13.12-c15-0-69
Degree $2$
Conductor $117$
Sign $0.743 + 0.668i$
Analytic cond. $166.951$
Root an. cond. $12.9209$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 234. i·2-s − 2.22e4·4-s − 9.94e4i·5-s − 2.97e6i·7-s + 2.46e6i·8-s + 2.33e7·10-s − 8.73e7i·11-s + (1.68e8 + 1.51e8i)13-s + 6.97e8·14-s − 1.30e9·16-s + 9.82e8·17-s − 4.85e9i·19-s + 2.21e9i·20-s + 2.04e10·22-s + 8.95e9·23-s + ⋯
L(s)  = 1  + 1.29i·2-s − 0.678·4-s − 0.569i·5-s − 1.36i·7-s + 0.416i·8-s + 0.737·10-s − 1.35i·11-s + (0.743 + 0.668i)13-s + 1.76·14-s − 1.21·16-s + 0.581·17-s − 1.24i·19-s + 0.386i·20-s + 1.75·22-s + 0.548·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.743 + 0.668i$
Analytic conductor: \(166.951\)
Root analytic conductor: \(12.9209\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :15/2),\ 0.743 + 0.668i)\)

Particular Values

\(L(8)\) \(\approx\) \(2.029900803\)
\(L(\frac12)\) \(\approx\) \(2.029900803\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-1.68e8 - 1.51e8i)T \)
good2 \( 1 - 234. iT - 3.27e4T^{2} \)
5 \( 1 + 9.94e4iT - 3.05e10T^{2} \)
7 \( 1 + 2.97e6iT - 4.74e12T^{2} \)
11 \( 1 + 8.73e7iT - 4.17e15T^{2} \)
17 \( 1 - 9.82e8T + 2.86e18T^{2} \)
19 \( 1 + 4.85e9iT - 1.51e19T^{2} \)
23 \( 1 - 8.95e9T + 2.66e20T^{2} \)
29 \( 1 + 1.53e10T + 8.62e21T^{2} \)
31 \( 1 - 2.37e11iT - 2.34e22T^{2} \)
37 \( 1 + 3.42e11iT - 3.33e23T^{2} \)
41 \( 1 + 8.57e10iT - 1.55e24T^{2} \)
43 \( 1 + 1.79e11T + 3.17e24T^{2} \)
47 \( 1 - 2.00e12iT - 1.20e25T^{2} \)
53 \( 1 - 4.75e12T + 7.31e25T^{2} \)
59 \( 1 + 3.16e13iT - 3.65e26T^{2} \)
61 \( 1 + 1.15e13T + 6.02e26T^{2} \)
67 \( 1 - 7.39e13iT - 2.46e27T^{2} \)
71 \( 1 + 3.85e13iT - 5.87e27T^{2} \)
73 \( 1 - 5.54e13iT - 8.90e27T^{2} \)
79 \( 1 - 9.17e13T + 2.91e28T^{2} \)
83 \( 1 + 2.48e14iT - 6.11e28T^{2} \)
89 \( 1 + 7.21e14iT - 1.74e29T^{2} \)
97 \( 1 + 1.48e15iT - 6.33e29T^{2} \)
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

−10.69838308609416691752283428048, −9.052594859188003769229338152505, −8.407854326676196884465368052923, −7.25867110564763152912346220458, −6.55154744562264832667716732635, −5.40057342011680107750262702840, −4.42473393860486506837484108909, −3.18453258350898696597237805116, −1.27613965080104009069867001308, −0.41277385783084053423064666538, 1.15805081978947951291201273205, 2.14691576985397800563721352192, 2.89469511319169375010018439354, 3.93223058657654576560533825426, 5.36892703295329119122015803725, 6.51305775741760916458062871091, 7.85902124695027815104272371393, 9.169567681462908733089787475390, 10.02537874753122455334391015952, 10.84233737229083136788207928909

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