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Properties

Label 2-117-117.5-c1-0-10
Degree $2$
Conductor $117$
Sign $-0.680 + 0.732i$
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  + (0.496 − 1.85i)2-s + (−0.584 − 1.63i)3-s + (−1.45 − 0.837i)4-s + (0.863 − 0.231i)5-s + (−3.30 + 0.274i)6-s + (−0.426 + 1.59i)7-s + (0.439 − 0.439i)8-s + (−2.31 + 1.90i)9-s − 1.71i·10-s + (0.152 + 0.0407i)11-s + (−0.517 + 2.85i)12-s + (−1.18 + 3.40i)13-s + (2.73 + 1.58i)14-s + (−0.882 − 1.27i)15-s + (−2.27 − 3.93i)16-s + 5.15·17-s + ⋯
L(s)  = 1  + (0.350 − 1.30i)2-s + (−0.337 − 0.941i)3-s + (−0.725 − 0.418i)4-s + (0.386 − 0.103i)5-s + (−1.35 + 0.112i)6-s + (−0.161 + 0.602i)7-s + (0.155 − 0.155i)8-s + (−0.771 + 0.635i)9-s − 0.542i·10-s + (0.0458 + 0.0122i)11-s + (−0.149 + 0.824i)12-s + (−0.328 + 0.944i)13-s + (0.731 + 0.422i)14-s + (−0.227 − 0.328i)15-s + (−0.567 − 0.983i)16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.680 + 0.732i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.680 + 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.472986 - 1.08475i\)
\(L(\frac12)\) \(\approx\) \(0.472986 - 1.08475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.584 + 1.63i)T \)
13 \( 1 + (1.18 - 3.40i)T \)
good2 \( 1 + (-0.496 + 1.85i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.863 + 0.231i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.426 - 1.59i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.152 - 0.0407i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 5.15T + 17T^{2} \)
19 \( 1 + (-1.74 + 1.74i)T - 19iT^{2} \)
23 \( 1 + (-1.40 + 2.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0404 - 0.0233i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.805 + 3.00i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.09 - 6.09i)T + 37iT^{2} \)
41 \( 1 + (8.85 - 2.37i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.93 - 5.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.53 + 0.678i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 9.99iT - 53T^{2} \)
59 \( 1 + (2.52 + 9.43i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.03 - 3.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.56 - 5.82i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (10.2 + 10.2i)T + 71iT^{2} \)
73 \( 1 + (10.2 + 10.2i)T + 73iT^{2} \)
79 \( 1 + (-3.37 - 5.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.35 + 12.5i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-7.53 + 7.53i)T - 89iT^{2} \)
97 \( 1 + (-10.7 - 2.89i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.00206556314337558184342420275, −11.86396849321383904451427853162, −11.66685267244919707750189206981, −10.22777323270711737845999895215, −9.203704321684732574979378399234, −7.62796968311427969992041485731, −6.29839960690888088999141514453, −4.92730733050517862983408303158, −2.97704252305839677333005213500, −1.63478317368172252957456941252, 3.62252640554648471720507168925, 5.15101829747769350982757654147, 5.85503455367265173405983099985, 7.17610839477391717535298712626, 8.299970631341868418191819083057, 9.813194433292435716397932621011, 10.50217200587544353472302285107, 11.86035650620868595600679733998, 13.30753999326083547670457601831, 14.28719120266149037222376899258

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