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Properties

Label 2-117-9.7-c1-0-6
Degree $2$
Conductor $117$
Sign $0.991 + 0.133i$
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  + (−1.39 + 2.42i)2-s + (0.413 − 1.68i)3-s + (−2.90 − 5.03i)4-s + (−0.413 − 0.717i)5-s + (3.49 + 3.35i)6-s + (1.24 − 2.15i)7-s + 10.6·8-s + (−2.65 − 1.39i)9-s + 2.31·10-s + (0.459 − 0.796i)11-s + (−9.67 + 2.80i)12-s + (−0.5 − 0.866i)13-s + (3.47 + 6.01i)14-s + (−1.37 + 0.399i)15-s + (−9.08 + 15.7i)16-s − 1.53·17-s + ⋯
L(s)  = 1  + (−0.988 + 1.71i)2-s + (0.239 − 0.971i)3-s + (−1.45 − 2.51i)4-s + (−0.185 − 0.320i)5-s + (1.42 + 1.36i)6-s + (0.469 − 0.813i)7-s + 3.76·8-s + (−0.885 − 0.464i)9-s + 0.731·10-s + (0.138 − 0.240i)11-s + (−2.79 + 0.809i)12-s + (−0.138 − 0.240i)13-s + (0.928 + 1.60i)14-s + (−0.355 + 0.103i)15-s + (−2.27 + 3.93i)16-s − 0.372·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.991 + 0.133i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.991 + 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.618227 - 0.0413731i\)
\(L(\frac12)\) \(\approx\) \(0.618227 - 0.0413731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.413 + 1.68i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.39 - 2.42i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.413 + 0.717i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.24 + 2.15i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.459 + 0.796i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 - 3.40T + 19T^{2} \)
23 \( 1 + (0.490 + 0.850i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.595 + 1.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.12 - 5.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.83T + 37T^{2} \)
41 \( 1 + (-2.86 - 4.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.792 - 1.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.68 - 4.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.82T + 53T^{2} \)
59 \( 1 + (-0.477 - 0.826i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.44 + 7.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.56 - 11.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.27T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 + (5.06 - 8.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.16 + 2.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + (-3.94 + 6.82i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.03712003881681358605990507094, −12.93643390873539059849961019663, −11.19949747110582826153274092743, −9.915154330202915091074058544069, −8.684877030031057036131710095507, −7.972609760099862616376290886388, −7.13162342158559728392625896078, −6.15119559422917816639441724312, −4.74503865279197543189741794207, −1.01232277694787782936942913152, 2.31882715963563022198515939422, 3.58507550871379443115494966123, 4.89681319362438311179293979568, 7.68389118924110958399288259982, 8.777382190259528145621716692589, 9.456040581122151296448786285857, 10.42048797130641020483316563391, 11.39699797267927088397137993667, 11.89757374288909569681091093284, 13.24954386025497884613330836545

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