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Properties

Label 2-117-117.4-c1-0-9
Degree $2$
Conductor $117$
Sign $-0.651 + 0.758i$
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  − 2.59i·2-s + (1.72 − 0.131i)3-s − 4.70·4-s + (1.18 + 0.685i)5-s + (−0.339 − 4.47i)6-s + (−3.20 − 1.85i)7-s + 7.01i·8-s + (2.96 − 0.453i)9-s + (1.77 − 3.07i)10-s + 0.487i·11-s + (−8.13 + 0.618i)12-s + (3.11 + 1.80i)13-s + (−4.79 + 8.31i)14-s + (2.13 + 1.02i)15-s + 8.76·16-s + (2.88 + 5.00i)17-s + ⋯
L(s)  = 1  − 1.83i·2-s + (0.997 − 0.0757i)3-s − 2.35·4-s + (0.530 + 0.306i)5-s + (−0.138 − 1.82i)6-s + (−1.21 − 0.700i)7-s + 2.48i·8-s + (0.988 − 0.151i)9-s + (0.561 − 0.972i)10-s + 0.146i·11-s + (−2.34 + 0.178i)12-s + (0.865 + 0.501i)13-s + (−1.28 + 2.22i)14-s + (0.552 + 0.265i)15-s + 2.19·16-s + (0.700 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.651 + 0.758i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.651 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513932 - 1.11955i\)
\(L(\frac12)\) \(\approx\) \(0.513932 - 1.11955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.131i)T \)
13 \( 1 + (-3.11 - 1.80i)T \)
good2 \( 1 + 2.59iT - 2T^{2} \)
5 \( 1 + (-1.18 - 0.685i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.20 + 1.85i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.487iT - 11T^{2} \)
17 \( 1 + (-2.88 - 5.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.80 - 1.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.175 + 0.304i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.00T + 29T^{2} \)
31 \( 1 + (3.62 + 2.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.56 + 4.36i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.44 + 0.836i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.78 - 10.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.26 + 1.88i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 5.70iT - 59T^{2} \)
61 \( 1 + (-1.17 + 2.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0679 + 0.0392i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.940 + 0.542i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.49iT - 73T^{2} \)
79 \( 1 + (1.16 + 2.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.19 - 1.26i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-14.0 - 8.08i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.732 + 0.423i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.01535636671520226402040785670, −12.40085622477377382309477003587, −10.77097641706690204408133315299, −10.10114314302745343612058874470, −9.386435180911511893334503012411, −8.251142323788899331992241814456, −6.42266006615703921064108244620, −4.06007758926594057587088655170, −3.29358103508902547306123576595, −1.79235792916729743780279327952, 3.36160161110781122742896581154, 5.14766869804087797047645427405, 6.24279738628132097811337889067, 7.29517472546396708183993743528, 8.578805124386371513407375017157, 9.146443490114556960633419296684, 10.03426446121249671841773851615, 12.55455021950447916459236798934, 13.42316018742738925245274615338, 13.95554817940635038499336770455

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