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Properties

Label 2-117-117.4-c1-0-6
Degree $2$
Conductor $117$
Sign $0.716 + 0.697i$
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  − 1.05i·2-s + (1.56 + 0.742i)3-s + 0.881·4-s + (−2.71 − 1.56i)5-s + (0.785 − 1.65i)6-s + (0.784 + 0.453i)7-s − 3.04i·8-s + (1.89 + 2.32i)9-s + (−1.65 + 2.86i)10-s + 5.46i·11-s + (1.37 + 0.654i)12-s + (−1.56 − 3.24i)13-s + (0.479 − 0.830i)14-s + (−3.07 − 4.46i)15-s − 1.46·16-s + (−1.52 − 2.63i)17-s + ⋯
L(s)  = 1  − 0.747i·2-s + (0.903 + 0.428i)3-s + 0.440·4-s + (−1.21 − 0.699i)5-s + (0.320 − 0.675i)6-s + (0.296 + 0.171i)7-s − 1.07i·8-s + (0.632 + 0.774i)9-s + (−0.523 + 0.906i)10-s + 1.64i·11-s + (0.398 + 0.188i)12-s + (−0.435 − 0.900i)13-s + (0.128 − 0.221i)14-s + (−0.795 − 1.15i)15-s − 0.365·16-s + (−0.369 − 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21697 - 0.494802i\)
\(L(\frac12)\) \(\approx\) \(1.21697 - 0.494802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.56 - 0.742i)T \)
13 \( 1 + (1.56 + 3.24i)T \)
good2 \( 1 + 1.05iT - 2T^{2} \)
5 \( 1 + (2.71 + 1.56i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.784 - 0.453i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.46iT - 11T^{2} \)
17 \( 1 + (1.52 + 2.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.69 - 3.29i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.18 - 3.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 + (1.19 + 0.687i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.84 - 1.06i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.42 + 4.28i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.83 - 3.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.01 + 2.89i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.37T + 53T^{2} \)
59 \( 1 - 3.14iT - 59T^{2} \)
61 \( 1 + (-4.09 + 7.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.7 + 6.77i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.07 + 1.77i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.25iT - 73T^{2} \)
79 \( 1 + (5.59 + 9.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.899 - 0.519i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.76 + 2.74i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.06 - 4.07i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.90853968642894547874544965884, −12.47520102063534029444987621781, −11.37974781963674735968510718565, −10.27940452415350089727269388873, −9.309868510897379909817003103606, −7.988651425032940341496737439085, −7.25717685273904289264937280479, −4.77966893049327107643648922591, −3.75647396566953185895902876684, −2.16050401759379877665024615265, 2.64802740380679121541540656317, 4.10533539325046862872939456706, 6.33878588969476549925390664705, 7.13000686743799612915709973461, 8.116768600937539420371197714227, 8.796097054958524801392064563109, 10.91833359974738524986790794597, 11.37817324541697360685851641859, 12.76168872655320074351281432804, 14.12274414968402127095332621287

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