Properties

Label 2-117-13.9-c1-0-3
Degree $2$
Conductor $117$
Sign $-0.396 + 0.918i$
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  + (1.28 − 2.21i)2-s + (−2.28 − 3.95i)4-s − 0.561·5-s + (1.78 + 3.08i)7-s − 6.56·8-s + (−0.719 + 1.24i)10-s + (−1 + 1.73i)11-s + (0.5 − 3.57i)13-s + 9.12·14-s + (−3.84 + 6.65i)16-s + (1.28 + 2.21i)17-s + (0.561 + 0.972i)19-s + (1.28 + 2.21i)20-s + (2.56 + 4.43i)22-s + (1 − 1.73i)23-s + ⋯
L(s)  = 1  + (0.905 − 1.56i)2-s + (−1.14 − 1.97i)4-s − 0.251·5-s + (0.673 + 1.16i)7-s − 2.31·8-s + (−0.227 + 0.393i)10-s + (−0.301 + 0.522i)11-s + (0.138 − 0.990i)13-s + 2.43·14-s + (−0.960 + 1.66i)16-s + (0.310 + 0.538i)17-s + (0.128 + 0.223i)19-s + (0.286 + 0.496i)20-s + (0.546 + 0.945i)22-s + (0.208 − 0.361i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.396 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819215 - 1.24568i\)
\(L(\frac12)\) \(\approx\) \(0.819215 - 1.24568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-0.5 + 3.57i)T \)
good2 \( 1 + (-1.28 + 2.21i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 0.561T + 5T^{2} \)
7 \( 1 + (-1.78 - 3.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.28 - 2.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.561 - 0.972i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.84 - 4.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 + (1.71 - 2.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.28 + 2.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.219 + 0.379i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (5.56 + 9.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.06 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.219 - 0.379i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7 - 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.87T + 73T^{2} \)
79 \( 1 - 9.56T + 79T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 + (-6.56 + 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.21 - 3.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.72210829843470083494383771382, −12.34412121947500947499630639408, −11.30148583256556690085665666628, −10.47841099184604268907636571723, −9.330856082135581568334057794199, −8.024620964287546698027762768591, −5.76694774139928555315823419937, −4.88834533012020725635976454799, −3.37501215095789122146119017909, −1.96676399542315194506728222206, 3.77826042602286638207070738084, 4.73990612066480177914652265560, 6.03745511979692414996077331983, 7.32437004952151628210810890975, 7.84732192698521126625977681321, 9.211622743321838273519581640093, 10.96653405830215826739740948575, 12.06946472649930519246685301197, 13.61198860505601370989456710363, 13.74494677279446661402001450339

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