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Properties

Label 2-117-117.32-c1-0-3
Degree $2$
Conductor $117$
Sign $0.678 + 0.734i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  + (−1.53 − 1.53i)2-s + (1.40 + 1.00i)3-s + 2.71i·4-s + (−0.529 + 0.142i)5-s + (−0.619 − 3.71i)6-s + (4.13 − 1.10i)7-s + (1.10 − 1.10i)8-s + (0.974 + 2.83i)9-s + (1.03 + 0.595i)10-s + (2.64 − 2.64i)11-s + (−2.73 + 3.83i)12-s + (−3.49 − 0.872i)13-s + (−8.06 − 4.65i)14-s + (−0.889 − 0.333i)15-s + 2.04·16-s + (−0.784 − 1.35i)17-s + ⋯
L(s)  = 1  + (−1.08 − 1.08i)2-s + (0.813 + 0.581i)3-s + 1.35i·4-s + (−0.237 + 0.0635i)5-s + (−0.252 − 1.51i)6-s + (1.56 − 0.419i)7-s + (0.389 − 0.389i)8-s + (0.324 + 0.945i)9-s + (0.326 + 0.188i)10-s + (0.798 − 0.798i)11-s + (−0.789 + 1.10i)12-s + (−0.970 − 0.242i)13-s + (−2.15 − 1.24i)14-s + (−0.229 − 0.0860i)15-s + 0.512·16-s + (−0.190 − 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.678 + 0.734i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778215 - 0.340557i\)
\(L(\frac12)\) \(\approx\) \(0.778215 - 0.340557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.40 - 1.00i)T \)
13 \( 1 + (3.49 + 0.872i)T \)
good2 \( 1 + (1.53 + 1.53i)T + 2iT^{2} \)
5 \( 1 + (0.529 - 0.142i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-4.13 + 1.10i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.64 + 2.64i)T - 11iT^{2} \)
17 \( 1 + (0.784 + 1.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.27 + 0.876i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.51 - 4.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.18iT - 29T^{2} \)
31 \( 1 + (-0.395 - 1.47i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.31 + 0.351i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.37 - 5.13i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.94 + 5.16i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.81 + 1.82i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 5.04iT - 53T^{2} \)
59 \( 1 + (7.65 - 7.65i)T - 59iT^{2} \)
61 \( 1 + (-2.96 + 5.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.07 + 0.824i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.774 + 2.89i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.10 + 9.10i)T + 73iT^{2} \)
79 \( 1 + (-7.18 - 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.67 + 13.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.63 - 6.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.846 - 3.15i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.40944706664091697988380616754, −11.75230014235551762159817397363, −11.18781987771255746124929762071, −10.25498700735143321132836274844, −9.197426063021747055906968166717, −8.353327163484507768697842653989, −7.53248875165629719819827787184, −4.89035273629691310133586006798, −3.42408714322963958336424617801, −1.79837603557507823888219416011, 1.87150160434170496134822466049, 4.54340107145309828383569111305, 6.39490757455415737710794210149, 7.44041985590458411990296141876, 8.212488847181904696611719568399, 8.934848277101174049164032828983, 10.01558120162293991685267058810, 11.67986184832326708543839009034, 12.59282068695968014774818317414, 14.30381050014522807643305574415

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