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Properties

Label 2-117-117.16-c1-0-3
Degree $2$
Conductor $117$
Sign $0.923 + 0.382i$
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  − 2.00·2-s + (−1.42 + 0.984i)3-s + 2.03·4-s + (−1.37 − 2.37i)5-s + (2.86 − 1.97i)6-s + (1.11 + 1.93i)7-s − 0.0636·8-s + (1.06 − 2.80i)9-s + (2.75 + 4.77i)10-s + 4.63·11-s + (−2.89 + 2.00i)12-s + (2.17 − 2.87i)13-s + (−2.24 − 3.89i)14-s + (4.29 + 2.03i)15-s − 3.93·16-s + (−0.925 + 1.60i)17-s + ⋯
L(s)  = 1  − 1.41·2-s + (−0.822 + 0.568i)3-s + 1.01·4-s + (−0.614 − 1.06i)5-s + (1.16 − 0.807i)6-s + (0.422 + 0.732i)7-s − 0.0224·8-s + (0.353 − 0.935i)9-s + (0.871 + 1.51i)10-s + 1.39·11-s + (−0.835 + 0.577i)12-s + (0.603 − 0.797i)13-s + (−0.600 − 1.04i)14-s + (1.11 + 0.525i)15-s − 0.983·16-s + (−0.224 + 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405820 - 0.0806800i\)
\(L(\frac12)\) \(\approx\) \(0.405820 - 0.0806800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.42 - 0.984i)T \)
13 \( 1 + (-2.17 + 2.87i)T \)
good2 \( 1 + 2.00T + 2T^{2} \)
5 \( 1 + (1.37 + 2.37i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.11 - 1.93i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.63T + 11T^{2} \)
17 \( 1 + (0.925 - 1.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.74 + 6.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.948 - 1.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.71T + 29T^{2} \)
31 \( 1 + (0.375 + 0.649i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.82 - 3.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.84 + 3.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.05 + 3.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.36 + 5.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.52T + 53T^{2} \)
59 \( 1 - 9.85T + 59T^{2} \)
61 \( 1 + (-4.78 - 8.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.12 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.04 - 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + (0.753 - 1.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.58 + 11.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.97 + 5.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.06 + 5.31i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.18682045418988607640189246889, −11.78111147657183108916032987399, −11.47695629105767245823573899341, −10.14947553466997288436760540481, −8.957307014228648981536146675794, −8.655522069518919897617672017068, −7.10135749875296911856949633502, −5.51969290531713986026701867392, −4.20951159672004688661348261889, −0.982881433596395107638976534375, 1.41032105884333390297141983552, 4.10554407165532527891774528289, 6.38320583389285603455745944892, 7.20362675611326475471196453817, 7.970945473265534781847477053808, 9.440734797803368788014745652744, 10.62422768771695744460937631344, 11.26635379114113499268969585630, 11.96316668344668277232609209630, 13.75667268647620050757903058380

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