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Properties

Label 2-117-117.88-c1-0-9
Degree $2$
Conductor $117$
Sign $0.202 + 0.979i$
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  − 0.571i·2-s + (−0.656 − 1.60i)3-s + 1.67·4-s + (0.796 − 0.459i)5-s + (−0.916 + 0.375i)6-s + (−1.67 + 0.966i)7-s − 2.10i·8-s + (−2.13 + 2.10i)9-s + (−0.262 − 0.455i)10-s − 4.20i·11-s + (−1.09 − 2.68i)12-s + (2.21 + 2.84i)13-s + (0.552 + 0.957i)14-s + (−1.25 − 0.974i)15-s + 2.14·16-s + (−1.20 + 2.08i)17-s + ⋯
L(s)  = 1  − 0.404i·2-s + (−0.378 − 0.925i)3-s + 0.836·4-s + (0.356 − 0.205i)5-s + (−0.374 + 0.153i)6-s + (−0.632 + 0.365i)7-s − 0.742i·8-s + (−0.713 + 0.701i)9-s + (−0.0831 − 0.143i)10-s − 1.26i·11-s + (−0.316 − 0.774i)12-s + (0.614 + 0.788i)13-s + (0.147 + 0.255i)14-s + (−0.325 − 0.251i)15-s + 0.536·16-s + (−0.291 + 0.505i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.202 + 0.979i$
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: \((2,\ 117,\ (\ :1/2),\ 0.202 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.856523 - 0.697177i\)
\(L(\frac12)\) \(\approx\) \(0.856523 - 0.697177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.656 + 1.60i)T \)
13 \( 1 + (-2.21 - 2.84i)T \)
good2 \( 1 + 0.571iT - 2T^{2} \)
5 \( 1 + (-0.796 + 0.459i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.67 - 0.966i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.20iT - 11T^{2} \)
17 \( 1 + (1.20 - 2.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.60 - 0.928i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.11 - 7.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 + (-4.29 + 2.47i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.959 - 0.554i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.490 + 0.283i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.79 + 8.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.35 - 0.780i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.09T + 53T^{2} \)
59 \( 1 - 6.11iT - 59T^{2} \)
61 \( 1 + (0.669 + 1.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (14.1 + 8.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.6 + 6.12i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.77iT - 73T^{2} \)
79 \( 1 + (-4.09 + 7.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.31 - 1.33i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (11.4 - 6.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.0 + 6.94i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.31274266115971592823551249876, −12.03908881015366119267088312666, −11.53687103526036613529722480067, −10.44505230450473892030782027920, −9.045096930511671961073946994922, −7.73756897775283822628797165191, −6.36755100451255364098501661942, −5.84900988679499471445733487252, −3.28721993147285926193233596026, −1.65398129225669043326014976600, 2.84734654230896545513281857690, 4.60500560289747727372221658582, 6.05544204845877911448733569559, 6.84958619650480365696147203172, 8.348635042168853849996053901888, 9.977312324354169173210289732991, 10.34329330821258399942679814679, 11.59311357543282266052981907131, 12.57409915471395292917979950611, 14.05218943262432037904555638897

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