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Properties

Label 2-117-9.4-c1-0-0
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: $\chi_{117} (40, \cdot )$
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

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

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