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Properties

Label 2-117-117.25-c1-0-5
Degree $2$
Conductor $117$
Sign $-0.276 + 0.961i$
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  + (−1.97 − 1.14i)2-s + (0.833 − 1.51i)3-s + (1.60 + 2.78i)4-s + (2.78 − 1.60i)5-s + (−3.38 + 2.05i)6-s + (2.09 + 1.21i)7-s − 2.76i·8-s + (−1.61 − 2.53i)9-s − 7.34·10-s + (−1.27 − 0.737i)11-s + (5.56 − 0.120i)12-s + (−2.24 + 2.82i)13-s + (−2.76 − 4.79i)14-s + (−0.121 − 5.56i)15-s + (0.0535 − 0.0927i)16-s − 5.12·17-s + ⋯
L(s)  = 1  + (−1.39 − 0.807i)2-s + (0.481 − 0.876i)3-s + (0.803 + 1.39i)4-s + (1.24 − 0.719i)5-s + (−1.38 + 0.837i)6-s + (0.793 + 0.457i)7-s − 0.978i·8-s + (−0.537 − 0.843i)9-s − 2.32·10-s + (−0.385 − 0.222i)11-s + (1.60 − 0.0348i)12-s + (−0.623 + 0.782i)13-s + (−0.739 − 1.28i)14-s + (−0.0312 − 1.43i)15-s + (0.0133 − 0.0231i)16-s − 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.276 + 0.961i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.276 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.455760 - 0.605371i\)
\(L(\frac12)\) \(\approx\) \(0.455760 - 0.605371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.833 + 1.51i)T \)
13 \( 1 + (2.24 - 2.82i)T \)
good2 \( 1 + (1.97 + 1.14i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.78 + 1.60i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.09 - 1.21i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.27 + 0.737i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.12T + 17T^{2} \)
19 \( 1 + 1.13iT - 19T^{2} \)
23 \( 1 + (-4.61 - 7.99i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.487 - 0.844i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.16 + 1.82i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.22iT - 37T^{2} \)
41 \( 1 + (3.47 - 2.00i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.33 + 0.769i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.739T + 53T^{2} \)
59 \( 1 + (-6.72 + 3.88i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.06 - 7.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.669 + 0.386i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.01iT - 71T^{2} \)
73 \( 1 - 9.21iT - 73T^{2} \)
79 \( 1 + (1.86 - 3.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.3 - 7.13i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.21iT - 89T^{2} \)
97 \( 1 + (13.1 + 7.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.14536762532089374193780357363, −11.96118697965701469827416266633, −11.17373869970823551406707407918, −9.673097766233157782888663556701, −9.034715346815634227816674331577, −8.296869140858360386429034661182, −7.00254325336706828992786977483, −5.31559350713646319647237069305, −2.45153059066128947207762353384, −1.54697697173793777268479535250, 2.40326758916929363980824957048, 4.88370949229018358946838910418, 6.32998103505861605584143294961, 7.54795956775400782130586723875, 8.592500619951351186832308367401, 9.553848833568419648421157178951, 10.50167473676063538950691584004, 10.77983567404075472533012619446, 13.16738949929936771128450708359, 14.40340412303121053282105863753

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