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Properties

Label 2-117-117.88-c1-0-5
Degree $2$
Conductor $117$
Sign $0.726 + 0.687i$
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  − 0.782i·2-s + (−1.67 + 0.449i)3-s + 1.38·4-s + (0.0536 − 0.0309i)5-s + (0.351 + 1.30i)6-s + (3.25 − 1.87i)7-s − 2.65i·8-s + (2.59 − 1.50i)9-s + (−0.0242 − 0.0419i)10-s + 1.28i·11-s + (−2.32 + 0.623i)12-s + (−3.60 − 0.171i)13-s + (−1.46 − 2.54i)14-s + (−0.0757 + 0.0758i)15-s + 0.699·16-s + (−2.74 + 4.75i)17-s + ⋯
L(s)  = 1  − 0.553i·2-s + (−0.965 + 0.259i)3-s + 0.693·4-s + (0.0239 − 0.0138i)5-s + (0.143 + 0.534i)6-s + (1.22 − 0.709i)7-s − 0.937i·8-s + (0.865 − 0.500i)9-s + (−0.00766 − 0.0132i)10-s + 0.386i·11-s + (−0.669 + 0.179i)12-s + (−0.998 − 0.0476i)13-s + (−0.392 − 0.680i)14-s + (−0.0195 + 0.0195i)15-s + 0.174·16-s + (−0.665 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943412 - 0.375545i\)
\(L(\frac12)\) \(\approx\) \(0.943412 - 0.375545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 - 0.449i)T \)
13 \( 1 + (3.60 + 0.171i)T \)
good2 \( 1 + 0.782iT - 2T^{2} \)
5 \( 1 + (-0.0536 + 0.0309i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.25 + 1.87i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.28iT - 11T^{2} \)
17 \( 1 + (2.74 - 4.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.72 - 1.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.82 + 4.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.13T + 29T^{2} \)
31 \( 1 + (3.62 - 2.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.03 - 2.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.56 - 4.94i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.11 + 3.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.73 + 1.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.752T + 53T^{2} \)
59 \( 1 + 0.359iT - 59T^{2} \)
61 \( 1 + (0.825 + 1.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.43 - 0.828i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.3 - 6.57i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.09iT - 73T^{2} \)
79 \( 1 + (0.616 - 1.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.0 - 6.38i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.30 + 0.751i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.68 - 5.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.99394815231234304836990673755, −12.15938642942375217064470322935, −11.14618204713779808585145318496, −10.70259307993297204573944013739, −9.627474921040673262805221316324, −7.70572422488786694235745726809, −6.79192874564790409938613485347, −5.27029102398924701782723208664, −4.01355691493998202918480034528, −1.68732234840674753399552724889, 2.15664367262363664942812943352, 4.98408383002387650979887655306, 5.68150061860204602525713547156, 7.09967864809409543196598309653, 7.80284910741378336326285668261, 9.371732672251460777721916597792, 11.07016201220328719322815990990, 11.44374580800976470759157208880, 12.31713014713646945378583160782, 13.79196236231644980592852715951

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