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Properties

Label 2-117-117.61-c1-0-5
Degree $2$
Conductor $117$
Sign $0.973 + 0.230i$
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  + (−0.348 − 0.604i)2-s + (1.58 + 0.690i)3-s + (0.756 − 1.31i)4-s + (1.44 + 2.50i)5-s + (−0.137 − 1.20i)6-s − 3.17·7-s − 2.45·8-s + (2.04 + 2.19i)9-s + (1.00 − 1.74i)10-s + (−1.15 − 2.00i)11-s + (2.10 − 1.56i)12-s + (0.0625 − 3.60i)13-s + (1.10 + 1.91i)14-s + (0.568 + 4.97i)15-s + (−0.658 − 1.14i)16-s + (2.69 + 4.66i)17-s + ⋯
L(s)  = 1  + (−0.246 − 0.427i)2-s + (0.917 + 0.398i)3-s + (0.378 − 0.655i)4-s + (0.646 + 1.11i)5-s + (−0.0560 − 0.490i)6-s − 1.19·7-s − 0.866·8-s + (0.682 + 0.730i)9-s + (0.318 − 0.552i)10-s + (−0.348 − 0.603i)11-s + (0.608 − 0.450i)12-s + (0.0173 − 0.999i)13-s + (0.295 + 0.512i)14-s + (0.146 + 1.28i)15-s + (−0.164 − 0.285i)16-s + (0.653 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24226 - 0.145013i\)
\(L(\frac12)\) \(\approx\) \(1.24226 - 0.145013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 - 0.690i)T \)
13 \( 1 + (-0.0625 + 3.60i)T \)
good2 \( 1 + (0.348 + 0.604i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.44 - 2.50i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.17T + 7T^{2} \)
11 \( 1 + (1.15 + 2.00i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.69 - 4.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.58 + 4.48i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.54T + 23T^{2} \)
29 \( 1 + (2.01 + 3.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.23 - 7.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.42 - 4.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.51T + 41T^{2} \)
43 \( 1 - 5.98T + 43T^{2} \)
47 \( 1 + (-0.521 + 0.902i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.29T + 53T^{2} \)
59 \( 1 + (-2.35 + 4.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 7.43T + 61T^{2} \)
67 \( 1 - 8.36T + 67T^{2} \)
71 \( 1 + (0.680 + 1.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + (0.0365 - 0.0633i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.08 - 1.88i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.0891 - 0.154i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.130T + 97T^{2} \)
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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

−13.65898664714798230900032874769, −12.61313607998642610040438751770, −10.85574865715042223582174017993, −10.24642831232279233951861184925, −9.707748105114677132221111546170, −8.319989975351187216212529314868, −6.71917175328271185030112959228, −5.80191443361525728467346675972, −3.35379003566676644401050250788, −2.47661041134812617400906153159, 2.29280565832374470147175838493, 3.93974676416283950122695365315, 6.00469387985743928490205132530, 7.13405073386642212742792243894, 8.174074344633497870865956365509, 9.287882609398140635557519894504, 9.766107675623918835674083560383, 12.07558565946831090667167853232, 12.61287798183247901644135490606, 13.43642101521820543699945340570

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