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Properties

Label 2-117-117.11-c1-0-11
Degree $2$
Conductor $117$
Sign $-0.334 + 0.942i$
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  + (1.75 − 1.75i)2-s + (−1.71 − 0.230i)3-s − 4.14i·4-s + (−0.542 − 0.145i)5-s + (−3.41 + 2.60i)6-s + (0.541 + 0.144i)7-s + (−3.76 − 3.76i)8-s + (2.89 + 0.790i)9-s + (−1.20 + 0.695i)10-s + (3.72 + 3.72i)11-s + (−0.954 + 7.11i)12-s + (−1.98 − 3.00i)13-s + (1.20 − 0.694i)14-s + (0.897 + 0.374i)15-s − 4.89·16-s + (−1.39 + 2.42i)17-s + ⋯
L(s)  = 1  + (1.23 − 1.23i)2-s + (−0.991 − 0.132i)3-s − 2.07i·4-s + (−0.242 − 0.0649i)5-s + (−1.39 + 1.06i)6-s + (0.204 + 0.0548i)7-s + (−1.33 − 1.33i)8-s + (0.964 + 0.263i)9-s + (−0.381 + 0.220i)10-s + (1.12 + 1.12i)11-s + (−0.275 + 2.05i)12-s + (−0.551 − 0.833i)13-s + (0.321 − 0.185i)14-s + (0.231 + 0.0966i)15-s − 1.22·16-s + (−0.339 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.334 + 0.942i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.334 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.829483 - 1.17506i\)
\(L(\frac12)\) \(\approx\) \(0.829483 - 1.17506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.230i)T \)
13 \( 1 + (1.98 + 3.00i)T \)
good2 \( 1 + (-1.75 + 1.75i)T - 2iT^{2} \)
5 \( 1 + (0.542 + 0.145i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.541 - 0.144i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.72 - 3.72i)T + 11iT^{2} \)
17 \( 1 + (1.39 - 2.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.31 + 0.888i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.77 - 6.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.895iT - 29T^{2} \)
31 \( 1 + (1.57 - 5.88i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.0565 + 0.0151i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.50 + 5.62i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.03 + 5.21i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.6 - 2.84i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 5.68iT - 53T^{2} \)
59 \( 1 + (-3.61 - 3.61i)T + 59iT^{2} \)
61 \( 1 + (-2.08 - 3.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.36 - 1.70i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.573 + 2.14i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.68 + 2.68i)T - 73iT^{2} \)
79 \( 1 + (-2.61 + 4.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.0375 - 0.139i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-2.14 + 7.99i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.238 - 0.888i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.84028501432569596028431181534, −12.07320427631109213944028338973, −11.62043384785452670697109552374, −10.45966050125281435103924950289, −9.645295919092598138348551977908, −7.39077814545992625511259119958, −5.94092281836995385137538293099, −4.88298267584568633370190108188, −3.82914969531675339274106637244, −1.72032583047790015929015760158, 3.81745336379555033649068493798, 4.80211670789593247920552744434, 6.04512037318534558053831305780, 6.74886949064069226863534211235, 7.936052376637265196644288053581, 9.511458197993654267626450501499, 11.37941894533098680640068030490, 11.83345319635196840458947689197, 12.99457272385663898374086719555, 14.08610786100480598136801671420

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