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Properties

Label 2-117-117.22-c1-0-0
Degree $2$
Conductor $117$
Sign $-0.952 - 0.304i$
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  − 2.28·2-s + (−0.409 + 1.68i)3-s + 3.21·4-s + (−0.461 + 0.799i)5-s + (0.935 − 3.84i)6-s + (−0.454 + 0.786i)7-s − 2.76·8-s + (−2.66 − 1.37i)9-s + (1.05 − 1.82i)10-s − 4.78·11-s + (−1.31 + 5.40i)12-s + (−2.54 + 2.54i)13-s + (1.03 − 1.79i)14-s + (−1.15 − 1.10i)15-s − 0.112·16-s + (1.98 + 3.44i)17-s + ⋯
L(s)  = 1  − 1.61·2-s + (−0.236 + 0.971i)3-s + 1.60·4-s + (−0.206 + 0.357i)5-s + (0.381 − 1.56i)6-s + (−0.171 + 0.297i)7-s − 0.977·8-s + (−0.888 − 0.459i)9-s + (0.333 − 0.576i)10-s − 1.44·11-s + (−0.379 + 1.55i)12-s + (−0.707 + 0.707i)13-s + (0.277 − 0.479i)14-s + (−0.298 − 0.285i)15-s − 0.0280·16-s + (0.481 + 0.834i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.952 - 0.304i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.952 - 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0388349 + 0.248767i\)
\(L(\frac12)\) \(\approx\) \(0.0388349 + 0.248767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.409 - 1.68i)T \)
13 \( 1 + (2.54 - 2.54i)T \)
good2 \( 1 + 2.28T + 2T^{2} \)
5 \( 1 + (0.461 - 0.799i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.454 - 0.786i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.78T + 11T^{2} \)
17 \( 1 + (-1.98 - 3.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.47 + 4.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.489 + 0.848i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.06T + 29T^{2} \)
31 \( 1 + (2.18 - 3.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.728 - 1.26i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.648 - 1.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.69 - 6.40i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.74 - 8.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 2.31T + 59T^{2} \)
61 \( 1 + (-6.26 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.780 - 1.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.58 - 7.93i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 + (4.86 + 8.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.11 + 14.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.29 - 3.97i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.09 - 7.09i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.39748144942388136245595544455, −12.64425768861313622023991358105, −11.28346415647624988836367331880, −10.61824974145615845154231691386, −9.813644027633977810323562252535, −8.858345806740065783742774480001, −7.85396496283906490546063474928, −6.53243042042171104954018010297, −4.86211635149944120754238452662, −2.75220844010313122783680381913, 0.44335391046638664084237195874, 2.44589095841391484067598140695, 5.36081485191582954317488477256, 6.95449861688529875077582421448, 7.84883190883476050827203271345, 8.435736871538288378407207748004, 9.982107901651616056358787906623, 10.65459936798260525731162241520, 11.94776304128397456229467402479, 12.79052994657381967614980134673

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