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Properties

Label 2-117-13.6-c2-0-8
Degree 22
Conductor 117117
Sign 0.884−0.466i0.884 - 0.466i
Analytic cond. 3.188013.18801
Root an. cond. 1.785501.78550
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 1.86i)2-s + (0.232 − 0.133i)4-s + (4.36 − 4.36i)5-s + (2.26 − 8.46i)7-s + (5.83 + 5.83i)8-s + (10.3 + 5.96i)10-s + (−6.19 + 1.66i)11-s + (−6.5 + 11.2i)13-s + 16.9·14-s + (−7.42 + 12.8i)16-s + (−9.99 + 5.76i)17-s + (3.36 + 0.901i)19-s + (0.428 − 1.59i)20-s + (−6.19 − 10.7i)22-s + (8.49 + 4.90i)23-s + ⋯
L(s)  = 1  + (0.250 + 0.933i)2-s + (0.0580 − 0.0334i)4-s + (0.873 − 0.873i)5-s + (0.323 − 1.20i)7-s + (0.728 + 0.728i)8-s + (1.03 + 0.596i)10-s + (−0.563 + 0.150i)11-s + (−0.5 + 0.866i)13-s + 1.20·14-s + (−0.464 + 0.804i)16-s + (−0.587 + 0.339i)17-s + (0.177 + 0.0474i)19-s + (0.0214 − 0.0799i)20-s + (−0.281 − 0.487i)22-s + (0.369 + 0.213i)23-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.884−0.466i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((0.884−0.466i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.884−0.466i0.884 - 0.466i
Analytic conductor: 3.188013.18801
Root analytic conductor: 1.785501.78550
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ117(19,⋅)\chi_{117} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1), 0.884−0.466i)(2,\ 117,\ (\ :1),\ 0.884 - 0.466i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.85976+0.460030i1.85976 + 0.460030i
L(12)L(\frac12) ≈\approx 1.85976+0.460030i1.85976 + 0.460030i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
13 1+(6.5−11.2i)T 1 + (6.5 - 11.2i)T
good2 1+(−0.5−1.86i)T+(−3.46+2i)T2 1 + (-0.5 - 1.86i)T + (-3.46 + 2i)T^{2}
5 1+(−4.36+4.36i)T−25iT2 1 + (-4.36 + 4.36i)T - 25iT^{2}
7 1+(−2.26+8.46i)T+(−42.4−24.5i)T2 1 + (-2.26 + 8.46i)T + (-42.4 - 24.5i)T^{2}
11 1+(6.19−1.66i)T+(104.−60.5i)T2 1 + (6.19 - 1.66i)T + (104. - 60.5i)T^{2}
17 1+(9.99−5.76i)T+(144.5−250.i)T2 1 + (9.99 - 5.76i)T + (144.5 - 250. i)T^{2}
19 1+(−3.36−0.901i)T+(312.+180.5i)T2 1 + (-3.36 - 0.901i)T + (312. + 180.5i)T^{2}
23 1+(−8.49−4.90i)T+(264.5+458.i)T2 1 + (-8.49 - 4.90i)T + (264.5 + 458. i)T^{2}
29 1+(5.69−9.86i)T+(−420.5−728.i)T2 1 + (5.69 - 9.86i)T + (-420.5 - 728. i)T^{2}
31 1+(−1.92+1.92i)T−961iT2 1 + (-1.92 + 1.92i)T - 961iT^{2}
37 1+(42.1−11.2i)T+(1.18e3−684.5i)T2 1 + (42.1 - 11.2i)T + (1.18e3 - 684.5i)T^{2}
41 1+(5.08+18.9i)T+(−1.45e3+840.5i)T2 1 + (5.08 + 18.9i)T + (-1.45e3 + 840.5i)T^{2}
43 1+(−45+25.9i)T+(924.5−1.60e3i)T2 1 + (-45 + 25.9i)T + (924.5 - 1.60e3i)T^{2}
47 1+(0.320+0.320i)T+2.20e3iT2 1 + (0.320 + 0.320i)T + 2.20e3iT^{2}
53 1+78.7T+2.80e3T2 1 + 78.7T + 2.80e3T^{2}
59 1+(10.9−40.9i)T+(−3.01e3−1.74e3i)T2 1 + (10.9 - 40.9i)T + (-3.01e3 - 1.74e3i)T^{2}
61 1+(49.1+85.1i)T+(−1.86e3+3.22e3i)T2 1 + (49.1 + 85.1i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(−19.9−74.5i)T+(−3.88e3+2.24e3i)T2 1 + (-19.9 - 74.5i)T + (-3.88e3 + 2.24e3i)T^{2}
71 1+(−31.0−8.31i)T+(4.36e3+2.52e3i)T2 1 + (-31.0 - 8.31i)T + (4.36e3 + 2.52e3i)T^{2}
73 1+(−48.2−48.2i)T+5.32e3iT2 1 + (-48.2 - 48.2i)T + 5.32e3iT^{2}
79 1+82.7T+6.24e3T2 1 + 82.7T + 6.24e3T^{2}
83 1+(−69.5+69.5i)T−6.88e3iT2 1 + (-69.5 + 69.5i)T - 6.88e3iT^{2}
89 1+(−31.8+8.52i)T+(6.85e3−3.96e3i)T2 1 + (-31.8 + 8.52i)T + (6.85e3 - 3.96e3i)T^{2}
97 1+(−74.8−20.0i)T+(8.14e3+4.70e3i)T2 1 + (-74.8 - 20.0i)T + (8.14e3 + 4.70e3i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−13.72476091100425639821470994967, −12.70089485289403244922890464251, −11.16260297646614537620239813661, −10.20251408990483604436756016377, −8.953550418840949620956976552447, −7.64452218802888765454604993187, −6.73221072594809962542770938333, −5.39693521225276615329407324784, −4.48601167121298465673405389167, −1.73552494048310208940632732094, 2.21734899732098843030985143830, 2.98660309311086061713674839472, 5.09527189466375724664663282836, 6.36730020975169760639902085143, 7.73971469795538567032374172500, 9.288510324567637433568674341859, 10.36647045301870142295350018769, 11.08966088582312802240054914470, 12.15886424512459801082979511783, 13.00663185896081728591841814558

Graph of the ZZ-function along the critical line