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Properties

Label 2-117-9.4-c1-0-3
Degree $2$
Conductor $117$
Sign $0.388 - 0.921i$
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  + (0.565 + 0.979i)2-s + (1.19 + 1.25i)3-s + (0.360 − 0.624i)4-s + (−1.19 + 2.06i)5-s + (−0.554 + 1.87i)6-s + (−2.04 − 3.54i)7-s + 3.07·8-s + (−0.149 + 2.99i)9-s − 2.70·10-s + (−1.92 − 3.33i)11-s + (1.21 − 0.292i)12-s + (−0.5 + 0.866i)13-s + (2.31 − 4.00i)14-s + (−4.01 + 0.970i)15-s + (1.01 + 1.76i)16-s + 2.15·17-s + ⋯
L(s)  = 1  + (0.399 + 0.692i)2-s + (0.689 + 0.724i)3-s + (0.180 − 0.312i)4-s + (−0.533 + 0.924i)5-s + (−0.226 + 0.767i)6-s + (−0.772 − 1.33i)7-s + 1.08·8-s + (−0.0499 + 0.998i)9-s − 0.853·10-s + (−0.581 − 1.00i)11-s + (0.350 − 0.0845i)12-s + (−0.138 + 0.240i)13-s + (0.617 − 1.07i)14-s + (−1.03 + 0.250i)15-s + (0.254 + 0.441i)16-s + 0.523·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.388 - 0.921i$
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: \((2,\ 117,\ (\ :1/2),\ 0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20059 + 0.796753i\)
\(L(\frac12)\) \(\approx\) \(1.20059 + 0.796753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.19 - 1.25i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.565 - 0.979i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.19 - 2.06i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.04 + 3.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.92 + 3.33i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.15T + 17T^{2} \)
19 \( 1 + 0.284T + 19T^{2} \)
23 \( 1 + (1.79 - 3.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.48 + 2.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.00 + 8.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.86T + 37T^{2} \)
41 \( 1 + (2.78 - 4.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.41 - 2.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.21 - 7.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.34T + 53T^{2} \)
59 \( 1 + (4.79 - 8.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.84 - 6.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.730 - 1.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.81T + 71T^{2} \)
73 \( 1 - 15.7T + 73T^{2} \)
79 \( 1 + (4.91 + 8.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.336 + 0.582i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.26T + 89T^{2} \)
97 \( 1 + (-4.43 - 7.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.82461329012809814745950757034, −13.44034850400346485926412636172, −11.25963938233948712272958441971, −10.51339781675344425995045450311, −9.771045064468843753270156652885, −7.920015170043019697824253442584, −7.21958962755411027917532750014, −5.93444429685444394633653072674, −4.27145296505379180507103362030, −3.17131766590844859478016914563, 2.16532857138919124091324421331, 3.38861773596318305848220549063, 5.04462449580866108894272511265, 6.85596191138548763001953194418, 8.078914324422530702775591120666, 8.877009343998280233403294274563, 10.20142032589294160548528309566, 12.03733289802740771003390434316, 12.40187216167828987238583658909, 12.77663521715212883287087366632

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