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Properties

Label 2-117-9.2-c2-0-6
Degree 22
Conductor 117117
Sign −0.775−0.631i-0.775 - 0.631i
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  + (2.73 + 1.58i)2-s + (−2.89 + 0.796i)3-s + (2.99 + 5.19i)4-s + (−6.90 + 3.98i)5-s + (−9.17 − 2.39i)6-s + (−1.88 + 3.27i)7-s + 6.30i·8-s + (7.73 − 4.60i)9-s − 25.2·10-s + (15.3 + 8.84i)11-s + (−12.8 − 12.6i)12-s + (1.80 + 3.12i)13-s + (−10.3 + 5.97i)14-s + (16.8 − 17.0i)15-s + (2.01 − 3.49i)16-s + 6.05i·17-s + ⋯
L(s)  = 1  + (1.36 + 0.790i)2-s + (−0.964 + 0.265i)3-s + (0.749 + 1.29i)4-s + (−1.38 + 0.797i)5-s + (−1.52 − 0.398i)6-s + (−0.269 + 0.467i)7-s + 0.788i·8-s + (0.858 − 0.512i)9-s − 2.52·10-s + (1.39 + 0.804i)11-s + (−1.06 − 1.05i)12-s + (0.138 + 0.240i)13-s + (−0.738 + 0.426i)14-s + (1.12 − 1.13i)15-s + (0.126 − 0.218i)16-s + 0.356i·17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((−0.775−0.631i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((−0.775−0.631i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: −0.775−0.631i-0.775 - 0.631i
Analytic conductor: 3.188013.18801
Root analytic conductor: 1.785501.78550
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ117(92,⋅)\chi_{117} (92, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1), −0.775−0.631i)(2,\ 117,\ (\ :1),\ -0.775 - 0.631i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.550510+1.54624i0.550510 + 1.54624i
L(12)L(\frac12) ≈\approx 0.550510+1.54624i0.550510 + 1.54624i
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+(2.89−0.796i)T 1 + (2.89 - 0.796i)T
13 1+(−1.80−3.12i)T 1 + (-1.80 - 3.12i)T
good2 1+(−2.73−1.58i)T+(2+3.46i)T2 1 + (-2.73 - 1.58i)T + (2 + 3.46i)T^{2}
5 1+(6.90−3.98i)T+(12.5−21.6i)T2 1 + (6.90 - 3.98i)T + (12.5 - 21.6i)T^{2}
7 1+(1.88−3.27i)T+(−24.5−42.4i)T2 1 + (1.88 - 3.27i)T + (-24.5 - 42.4i)T^{2}
11 1+(−15.3−8.84i)T+(60.5+104.i)T2 1 + (-15.3 - 8.84i)T + (60.5 + 104. i)T^{2}
17 1−6.05iT−289T2 1 - 6.05iT - 289T^{2}
19 1−13.2T+361T2 1 - 13.2T + 361T^{2}
23 1+(24.8−14.3i)T+(264.5−458.i)T2 1 + (24.8 - 14.3i)T + (264.5 - 458. i)T^{2}
29 1+(−18.0−10.4i)T+(420.5+728.i)T2 1 + (-18.0 - 10.4i)T + (420.5 + 728. i)T^{2}
31 1+(21.5+37.2i)T+(−480.5+832.i)T2 1 + (21.5 + 37.2i)T + (-480.5 + 832. i)T^{2}
37 1−1.44T+1.36e3T2 1 - 1.44T + 1.36e3T^{2}
41 1+(13.0−7.55i)T+(840.5−1.45e3i)T2 1 + (13.0 - 7.55i)T + (840.5 - 1.45e3i)T^{2}
43 1+(−21.7+37.6i)T+(−924.5−1.60e3i)T2 1 + (-21.7 + 37.6i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(−34.1−19.7i)T+(1.10e3+1.91e3i)T2 1 + (-34.1 - 19.7i)T + (1.10e3 + 1.91e3i)T^{2}
53 1−77.4iT−2.80e3T2 1 - 77.4iT - 2.80e3T^{2}
59 1+(−79.5+45.9i)T+(1.74e3−3.01e3i)T2 1 + (-79.5 + 45.9i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(−12.3+21.4i)T+(−1.86e3−3.22e3i)T2 1 + (-12.3 + 21.4i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(63.4+109.i)T+(−2.24e3+3.88e3i)T2 1 + (63.4 + 109. i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1+79.9iT−5.04e3T2 1 + 79.9iT - 5.04e3T^{2}
73 1−15.5T+5.32e3T2 1 - 15.5T + 5.32e3T^{2}
79 1+(50.7−87.9i)T+(−3.12e3−5.40e3i)T2 1 + (50.7 - 87.9i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−11.0−6.38i)T+(3.44e3+5.96e3i)T2 1 + (-11.0 - 6.38i)T + (3.44e3 + 5.96e3i)T^{2}
89 1−127.iT−7.92e3T2 1 - 127. iT - 7.92e3T^{2}
97 1+(−13.3+23.1i)T+(−4.70e3−8.14e3i)T2 1 + (-13.3 + 23.1i)T + (-4.70e3 - 8.14e3i)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

−13.92842535145016567514886880899, −12.26229178351239554618603004240, −12.10384978769465358965680144769, −11.10326807782303440323387887359, −9.566775760342048497191516765682, −7.56860590809157225856054059862, −6.77660416126717208033120814367, −5.85199427373986461253086149018, −4.33212105456582657684338852889, −3.66886604993294519558202294254, 0.944237323125544040478075302942, 3.66340202815230792171238206527, 4.42645269970873825266412866247, 5.67054207039197744346902095299, 6.97871357361311155095446160961, 8.483482605916413829736180486136, 10.32442016109587989941354367696, 11.58001575817014546822734064670, 11.76501267831768257687956514148, 12.63420729080294916369187948071

Graph of the ZZ-function along the critical line