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Properties

Label 2-117-117.88-c1-0-8
Degree $2$
Conductor $117$
Sign $0.107 + 0.994i$
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  − 2.00i·2-s + (1.62 + 0.587i)3-s − 2.02·4-s + (0.778 − 0.449i)5-s + (1.17 − 3.26i)6-s + (−2.10 + 1.21i)7-s + 0.0495i·8-s + (2.30 + 1.91i)9-s + (−0.901 − 1.56i)10-s − 0.984i·11-s + (−3.29 − 1.19i)12-s + (−3.22 + 1.61i)13-s + (2.44 + 4.23i)14-s + (1.53 − 0.274i)15-s − 3.94·16-s + (−2.01 + 3.48i)17-s + ⋯
L(s)  = 1  − 1.41i·2-s + (0.940 + 0.339i)3-s − 1.01·4-s + (0.348 − 0.200i)5-s + (0.481 − 1.33i)6-s + (−0.797 + 0.460i)7-s + 0.0175i·8-s + (0.769 + 0.638i)9-s + (−0.285 − 0.493i)10-s − 0.296i·11-s + (−0.952 − 0.343i)12-s + (−0.894 + 0.447i)13-s + (0.652 + 1.13i)14-s + (0.395 − 0.0709i)15-s − 0.987·16-s + (−0.488 + 0.845i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.107 + 0.994i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.107 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.977036 - 0.877090i\)
\(L(\frac12)\) \(\approx\) \(0.977036 - 0.877090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 - 0.587i)T \)
13 \( 1 + (3.22 - 1.61i)T \)
good2 \( 1 + 2.00iT - 2T^{2} \)
5 \( 1 + (-0.778 + 0.449i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.10 - 1.21i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.984iT - 11T^{2} \)
17 \( 1 + (2.01 - 3.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.316 - 0.182i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.55 + 4.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.88T + 29T^{2} \)
31 \( 1 + (-5.63 + 3.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.80 + 2.19i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.50 + 4.90i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.11 - 5.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.09 + 4.67i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.76T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + (7.41 + 12.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.95 - 2.28i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.72 - 2.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 16.9iT - 73T^{2} \)
79 \( 1 + (-5.57 + 9.64i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.50 + 0.866i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.61 + 2.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.54 - 1.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.03884834158194512871631952601, −12.42491849039480271434811462457, −11.13288586940623721732140796342, −9.976827665545931479886553902201, −9.465686354949528991352700940867, −8.412078644485032798061452677757, −6.62517111427107371184611872663, −4.61603864939442601133416255005, −3.25403297117828302119991093442, −2.14327506152334770359802910913, 2.82237875507951473597028751513, 4.75947498918422474319278078458, 6.42246260683959253476327913579, 7.13081212454007757753204451760, 8.061651038989324308660388994301, 9.292885360555585030817521310905, 10.12335753602668430498013203422, 12.03744654628907991311547816442, 13.40215423401378131480522929803, 13.81868993521487114540835523193

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