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Properties

Label 2-117-9.4-c1-0-5
Degree $2$
Conductor $117$
Sign $0.953 - 0.302i$
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  + (0.786 + 1.36i)2-s + (−0.557 − 1.63i)3-s + (−0.236 + 0.409i)4-s + (0.557 − 0.966i)5-s + (1.79 − 2.04i)6-s + (1.93 + 3.35i)7-s + 2.40·8-s + (−2.37 + 1.82i)9-s + 1.75·10-s + (−2.48 − 4.31i)11-s + (0.802 + 0.159i)12-s + (−0.5 + 0.866i)13-s + (−3.04 + 5.27i)14-s + (−1.89 − 0.375i)15-s + (2.36 + 4.08i)16-s − 4.37·17-s + ⋯
L(s)  = 1  + (0.555 + 0.962i)2-s + (−0.322 − 0.946i)3-s + (−0.118 + 0.204i)4-s + (0.249 − 0.432i)5-s + (0.732 − 0.836i)6-s + (0.731 + 1.26i)7-s + 0.849·8-s + (−0.792 + 0.609i)9-s + 0.554·10-s + (−0.750 − 1.30i)11-s + (0.231 + 0.0459i)12-s + (−0.138 + 0.240i)13-s + (−0.813 + 1.40i)14-s + (−0.489 − 0.0969i)15-s + (0.590 + 1.02i)16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.953 - 0.302i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.953 - 0.302i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33499 + 0.206483i\)
\(L(\frac12)\) \(\approx\) \(1.33499 + 0.206483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.557 + 1.63i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.786 - 1.36i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.557 + 0.966i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.93 - 3.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.48 + 4.31i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 4.37T + 17T^{2} \)
19 \( 1 + 6.30T + 19T^{2} \)
23 \( 1 + (0.977 - 1.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.08 - 1.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 + 2.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.10T + 37T^{2} \)
41 \( 1 + (2.75 - 4.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.94 + 6.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.99 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.68T + 53T^{2} \)
59 \( 1 + (-2.34 + 4.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.06 + 1.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.16 - 2.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.31T + 71T^{2} \)
73 \( 1 - 7.66T + 73T^{2} \)
79 \( 1 + (-1.49 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.98 + 12.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.98T + 89T^{2} \)
97 \( 1 + (3.82 + 6.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.47302879859778999400830724769, −12.98953821513573346029314255616, −11.64182237702486273039963773216, −10.82128566945930231010788062944, −8.702811403868188080986871502063, −8.111703979143738017985702828358, −6.63051075195380412370956004726, −5.74218857039768356832713785665, −4.96553661032141410860719389960, −2.11576874260216266956143852443, 2.45091999925774582283453424496, 4.23462424113143446272374523413, 4.70785704976453190994270730966, 6.74962584606183136145544276223, 8.096785376104754985271857291184, 9.954796297873403048279484160006, 10.59566816188504968326069237097, 11.11082156932817837828174506649, 12.37977175177168980092268085252, 13.35147008760769711878013880133

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