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Properties

Label 2-117-9.4-c1-0-9
Degree $2$
Conductor $117$
Sign $0.293 + 0.955i$
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  + (−0.200 − 0.346i)2-s + (0.771 − 1.55i)3-s + (0.919 − 1.59i)4-s + (−0.771 + 1.33i)5-s + (−0.691 + 0.0427i)6-s + (0.0370 + 0.0641i)7-s − 1.53·8-s + (−1.80 − 2.39i)9-s + 0.617·10-s + (1.40 + 2.42i)11-s + (−1.76 − 2.65i)12-s + (−0.5 + 0.866i)13-s + (0.0148 − 0.0256i)14-s + (1.47 + 2.22i)15-s + (−1.53 − 2.65i)16-s + 3.02·17-s + ⋯
L(s)  = 1  + (−0.141 − 0.245i)2-s + (0.445 − 0.895i)3-s + (0.459 − 0.796i)4-s + (−0.345 + 0.597i)5-s + (−0.282 + 0.0174i)6-s + (0.0139 + 0.0242i)7-s − 0.543·8-s + (−0.602 − 0.797i)9-s + 0.195·10-s + (0.422 + 0.731i)11-s + (−0.508 − 0.766i)12-s + (−0.138 + 0.240i)13-s + (0.00396 − 0.00686i)14-s + (0.381 + 0.575i)15-s + (−0.383 − 0.663i)16-s + 0.733·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.293 + 0.955i$
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: \((2,\ 117,\ (\ :1/2),\ 0.293 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914189 - 0.675476i\)
\(L(\frac12)\) \(\approx\) \(0.914189 - 0.675476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.771 + 1.55i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.200 + 0.346i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.771 - 1.33i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.0370 - 0.0641i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.40 - 2.42i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.02T + 17T^{2} \)
19 \( 1 - 4.58T + 19T^{2} \)
23 \( 1 + (3.12 - 5.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.39 + 4.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.70 - 4.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.01T + 37T^{2} \)
41 \( 1 + (0.0193 - 0.0334i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.98 - 6.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.82 + 11.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.37T + 53T^{2} \)
59 \( 1 + (-0.694 + 1.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.85 + 8.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.06 - 7.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.37T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + (2.66 + 4.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.41 + 12.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + (-5.60 - 9.71i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.41884398112752192565585207771, −11.97638558440991549528844588565, −11.55714859297125020130233133704, −10.11790673018268743887631783354, −9.210776177175775123662237445461, −7.58904850149260723506816827485, −6.89080479020364660584413361240, −5.59977413898723846282630773244, −3.30335799799393020935783278623, −1.70357110522679958396523609609, 3.02357838036202714987798752621, 4.20889482784192499237341482688, 5.77629636229096381399223864661, 7.52053267393612957057123825577, 8.416324055031818116941455184292, 9.245748418824993763761140868468, 10.61174042311276106484316931313, 11.70388770521619353298649662459, 12.58268171077473184536776388544, 13.91321896308989922975457015552

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