↑

Properties

Label 2-117-13.7-c2-0-7
Degree $2$
Conductor $117$
Sign $0.964 + 0.264i$
Analytic cond. $3.18801$
Root an. cond. $1.78550$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.06 − 0.822i)2-s + (5.27 − 3.04i)4-s + (5.58 + 5.58i)5-s + (−7.30 − 1.95i)7-s + (4.69 − 4.69i)8-s + (21.7 + 12.5i)10-s + (−3.01 − 11.2i)11-s + (−12.4 − 3.76i)13-s − 24.0·14-s + (−1.64 + 2.84i)16-s + (10.4 − 6.05i)17-s + (2.73 − 10.1i)19-s + (46.4 + 12.4i)20-s + (−18.4 − 32.0i)22-s + (3.75 + 2.16i)23-s + ⋯
L(s)  = 1  + (1.53 − 0.411i)2-s + (1.31 − 0.761i)4-s + (1.11 + 1.11i)5-s + (−1.04 − 0.279i)7-s + (0.586 − 0.586i)8-s + (2.17 + 1.25i)10-s + (−0.273 − 1.02i)11-s + (−0.957 − 0.289i)13-s − 1.71·14-s + (−0.102 + 0.177i)16-s + (0.616 − 0.355i)17-s + (0.143 − 0.536i)19-s + (2.32 + 0.622i)20-s + (−0.840 − 1.45i)22-s + (0.163 + 0.0942i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.964 + 0.264i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ 0.964 + 0.264i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.98845 - 0.402942i\)
\(L(\frac12)\) \(\approx\) \(2.98845 - 0.402942i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (12.4 + 3.76i)T \)
good2 \( 1 + (-3.06 + 0.822i)T + (3.46 - 2i)T^{2} \)
5 \( 1 + (-5.58 - 5.58i)T + 25iT^{2} \)
7 \( 1 + (7.30 + 1.95i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (3.01 + 11.2i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-10.4 + 6.05i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.73 + 10.1i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-3.75 - 2.16i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (22.1 - 38.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (5.71 + 5.71i)T + 961iT^{2} \)
37 \( 1 + (-8.51 - 31.7i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-58.1 + 15.5i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-27.0 + 15.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (22.4 - 22.4i)T - 2.20e3iT^{2} \)
53 \( 1 - 54.7T + 2.80e3T^{2} \)
59 \( 1 + (-13.4 - 3.61i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (42.0 + 72.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-62.2 + 16.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-14.8 + 55.2i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (54.9 - 54.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 45.0T + 6.24e3T^{2} \)
83 \( 1 + (-30.8 - 30.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (5.84 + 21.8i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-28.7 + 107. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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

−13.32320845035805097652694742830, −12.59868268691812867175753678674, −11.24106084971998889082682210719, −10.43000430281335636497624081976, −9.417201797359041677340228281251, −7.15773741436931077895895141280, −6.16485215216766736646164623872, −5.29960695127949138926461678966, −3.37312451787555492692349172075, −2.63424611563544666976536831001, 2.39878544364948873692355449719, 4.22710528124670227309935638839, 5.38019209858196876565615055410, 6.10823049673208701997054505840, 7.44618534116947661133842645631, 9.334018822123375804783373238187, 9.931503867446823317085929500969, 12.00966329077339014054747601190, 12.75102435290159903092945112886, 13.10217236651893767145714461385

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