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Properties

Label 2-117-117.4-c1-0-1
Degree $2$
Conductor $117$
Sign $-0.716 - 0.697i$
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  + 1.93i·2-s + (0.820 + 1.52i)3-s − 1.74·4-s + (−2.26 − 1.30i)5-s + (−2.95 + 1.58i)6-s + (2.01 + 1.16i)7-s + 0.497i·8-s + (−1.65 + 2.50i)9-s + (2.53 − 4.38i)10-s − 5.38i·11-s + (−1.42 − 2.65i)12-s + (3.56 − 0.534i)13-s + (−2.25 + 3.90i)14-s + (0.137 − 4.53i)15-s − 4.44·16-s + (0.835 + 1.44i)17-s + ⋯
L(s)  = 1  + 1.36i·2-s + (0.473 + 0.880i)3-s − 0.871·4-s + (−1.01 − 0.585i)5-s + (−1.20 + 0.647i)6-s + (0.762 + 0.440i)7-s + 0.175i·8-s + (−0.551 + 0.834i)9-s + (0.800 − 1.38i)10-s − 1.62i·11-s + (−0.412 − 0.767i)12-s + (0.988 − 0.148i)13-s + (−0.602 + 1.04i)14-s + (0.0354 − 1.17i)15-s − 1.11·16-s + (0.202 + 0.351i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.716 - 0.697i$
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: \((2,\ 117,\ (\ :1/2),\ -0.716 - 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420497 + 1.03539i\)
\(L(\frac12)\) \(\approx\) \(0.420497 + 1.03539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.820 - 1.52i)T \)
13 \( 1 + (-3.56 + 0.534i)T \)
good2 \( 1 - 1.93iT - 2T^{2} \)
5 \( 1 + (2.26 + 1.30i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.01 - 1.16i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.38iT - 11T^{2} \)
17 \( 1 + (-0.835 - 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.90 + 1.09i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.10 - 3.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.72T + 29T^{2} \)
31 \( 1 + (7.55 + 4.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.22 + 0.707i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.31 - 0.761i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.938 - 1.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.47 + 2.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.81T + 53T^{2} \)
59 \( 1 + 7.61iT - 59T^{2} \)
61 \( 1 + (-0.467 + 0.809i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.79 - 1.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.9 - 6.33i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 + (-3.46 - 6.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.95 - 4.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.62 - 4.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.03 - 1.17i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.27685008930476241231959121968, −13.41124433862313507305526056297, −11.56679659994619112087394518698, −10.99430402059996499724373534893, −9.005445125797873381527119257129, −8.392893426985889741415574620027, −7.79185168780298599663993297548, −5.90580901256531144521120493496, −4.96497680204054662307443724213, −3.57076572943277634384150537574, 1.56308506352440672837420137198, 3.13782012051928188666974616392, 4.35637478287072612701075454396, 6.89383799970305869488631241534, 7.63004083919591543897164538249, 8.955831130145897635386163593122, 10.35547413949010428826505274811, 11.27806704144543700468513042247, 12.05098867587702986919566930737, 12.77316570887976375812047907245

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