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Properties

Label 2-117-9.2-c2-0-11
Degree 22
Conductor 117117
Sign 0.613−0.789i0.613 - 0.789i
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  + (3.19 + 1.84i)2-s + (−2.62 − 1.45i)3-s + (4.78 + 8.28i)4-s + (7.14 − 4.12i)5-s + (−5.70 − 9.46i)6-s + (−2.39 + 4.15i)7-s + 20.5i·8-s + (4.78 + 7.62i)9-s + 30.4·10-s + (−12.2 − 7.06i)11-s + (−0.536 − 28.7i)12-s + (1.80 + 3.12i)13-s + (−15.2 + 8.83i)14-s + (−24.7 + 0.462i)15-s + (−18.6 + 32.2i)16-s − 19.3i·17-s + ⋯
L(s)  = 1  + (1.59 + 0.920i)2-s + (−0.875 − 0.483i)3-s + (1.19 + 2.07i)4-s + (1.42 − 0.825i)5-s + (−0.950 − 1.57i)6-s + (−0.342 + 0.593i)7-s + 2.56i·8-s + (0.532 + 0.846i)9-s + 3.04·10-s + (−1.11 − 0.641i)11-s + (−0.0447 − 2.39i)12-s + (0.138 + 0.240i)13-s + (−1.09 + 0.630i)14-s + (−1.65 + 0.0308i)15-s + (−1.16 + 2.01i)16-s − 1.14i·17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.613−0.789i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((0.613−0.789i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.613−0.789i0.613 - 0.789i
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), 0.613−0.789i)(2,\ 117,\ (\ :1),\ 0.613 - 0.789i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.47701+1.21191i2.47701 + 1.21191i
L(12)L(\frac12) ≈\approx 2.47701+1.21191i2.47701 + 1.21191i
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+(2.62+1.45i)T 1 + (2.62 + 1.45i)T
13 1+(−1.80−3.12i)T 1 + (-1.80 - 3.12i)T
good2 1+(−3.19−1.84i)T+(2+3.46i)T2 1 + (-3.19 - 1.84i)T + (2 + 3.46i)T^{2}
5 1+(−7.14+4.12i)T+(12.5−21.6i)T2 1 + (-7.14 + 4.12i)T + (12.5 - 21.6i)T^{2}
7 1+(2.39−4.15i)T+(−24.5−42.4i)T2 1 + (2.39 - 4.15i)T + (-24.5 - 42.4i)T^{2}
11 1+(12.2+7.06i)T+(60.5+104.i)T2 1 + (12.2 + 7.06i)T + (60.5 + 104. i)T^{2}
17 1+19.3iT−289T2 1 + 19.3iT - 289T^{2}
19 1+12.4T+361T2 1 + 12.4T + 361T^{2}
23 1+(6.83−3.94i)T+(264.5−458.i)T2 1 + (6.83 - 3.94i)T + (264.5 - 458. i)T^{2}
29 1+(0.567+0.327i)T+(420.5+728.i)T2 1 + (0.567 + 0.327i)T + (420.5 + 728. i)T^{2}
31 1+(1.46+2.53i)T+(−480.5+832.i)T2 1 + (1.46 + 2.53i)T + (-480.5 + 832. i)T^{2}
37 1+65.5T+1.36e3T2 1 + 65.5T + 1.36e3T^{2}
41 1+(−24.7+14.2i)T+(840.5−1.45e3i)T2 1 + (-24.7 + 14.2i)T + (840.5 - 1.45e3i)T^{2}
43 1+(−7.83+13.5i)T+(−924.5−1.60e3i)T2 1 + (-7.83 + 13.5i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(−59.3−34.2i)T+(1.10e3+1.91e3i)T2 1 + (-59.3 - 34.2i)T + (1.10e3 + 1.91e3i)T^{2}
53 1−54.1iT−2.80e3T2 1 - 54.1iT - 2.80e3T^{2}
59 1+(41.3−23.8i)T+(1.74e3−3.01e3i)T2 1 + (41.3 - 23.8i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(40.5−70.3i)T+(−1.86e3−3.22e3i)T2 1 + (40.5 - 70.3i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−26.6−46.1i)T+(−2.24e3+3.88e3i)T2 1 + (-26.6 - 46.1i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1+41.0iT−5.04e3T2 1 + 41.0iT - 5.04e3T^{2}
73 1+12.7T+5.32e3T2 1 + 12.7T + 5.32e3T^{2}
79 1+(−67.1+116.i)T+(−3.12e3−5.40e3i)T2 1 + (-67.1 + 116. i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−86.4−49.8i)T+(3.44e3+5.96e3i)T2 1 + (-86.4 - 49.8i)T + (3.44e3 + 5.96e3i)T^{2}
89 1+10.4iT−7.92e3T2 1 + 10.4iT - 7.92e3T^{2}
97 1+(−70.6+122.i)T+(−4.70e3−8.14e3i)T2 1 + (-70.6 + 122. i)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

−13.57790956701977958987035684570, −12.67644871701404400035993345010, −12.06707090581338709092370427163, −10.61858002471280347548332636818, −8.922431874358970871432335333713, −7.43201530665177185615547818468, −6.11212658846009120777546521445, −5.63498821929614450671684935079, −4.78672274858676219363269240243, −2.46779780707619275535103911354, 2.07283935319548868561533993179, 3.62498583981416343557106342786, 5.03276309343149442699892645269, 5.96425788826780249619441822238, 6.76994141582970310781312134700, 9.836180272944491290950938796104, 10.52590251007965626741888230188, 10.79795523484610385892186232628, 12.39366848634645656215029709311, 13.03762254269407731657641650160

Graph of the ZZ-function along the critical line