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Properties

Label 2-117-9.2-c2-0-12
Degree 22
Conductor 117117
Sign 0.752+0.658i0.752 + 0.658i
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  + (1.10 + 0.635i)2-s + (−2.90 + 0.746i)3-s + (−1.19 − 2.06i)4-s + (4.29 − 2.47i)5-s + (−3.67 − 1.02i)6-s + (2.07 − 3.59i)7-s − 8.11i·8-s + (7.88 − 4.34i)9-s + 6.30·10-s + (6.91 + 3.99i)11-s + (5.00 + 5.10i)12-s + (−1.80 − 3.12i)13-s + (4.57 − 2.64i)14-s + (−10.6 + 10.4i)15-s + (0.388 − 0.673i)16-s − 19.5i·17-s + ⋯
L(s)  = 1  + (0.550 + 0.317i)2-s + (−0.968 + 0.248i)3-s + (−0.298 − 0.516i)4-s + (0.858 − 0.495i)5-s + (−0.612 − 0.170i)6-s + (0.296 − 0.514i)7-s − 1.01i·8-s + (0.876 − 0.482i)9-s + 0.630·10-s + (0.628 + 0.362i)11-s + (0.417 + 0.425i)12-s + (−0.138 − 0.240i)13-s + (0.326 − 0.188i)14-s + (−0.708 + 0.693i)15-s + (0.0242 − 0.0420i)16-s − 1.14i·17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.752+0.658i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((0.752+0.658i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.752+0.658i0.752 + 0.658i
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), 0.752+0.658i)(2,\ 117,\ (\ :1),\ 0.752 + 0.658i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.34735−0.505962i1.34735 - 0.505962i
L(12)L(\frac12) ≈\approx 1.34735−0.505962i1.34735 - 0.505962i
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+(2.90−0.746i)T 1 + (2.90 - 0.746i)T
13 1+(1.80+3.12i)T 1 + (1.80 + 3.12i)T
good2 1+(−1.10−0.635i)T+(2+3.46i)T2 1 + (-1.10 - 0.635i)T + (2 + 3.46i)T^{2}
5 1+(−4.29+2.47i)T+(12.5−21.6i)T2 1 + (-4.29 + 2.47i)T + (12.5 - 21.6i)T^{2}
7 1+(−2.07+3.59i)T+(−24.5−42.4i)T2 1 + (-2.07 + 3.59i)T + (-24.5 - 42.4i)T^{2}
11 1+(−6.91−3.99i)T+(60.5+104.i)T2 1 + (-6.91 - 3.99i)T + (60.5 + 104. i)T^{2}
17 1+19.5iT−289T2 1 + 19.5iT - 289T^{2}
19 1−5.83T+361T2 1 - 5.83T + 361T^{2}
23 1+(8.95−5.17i)T+(264.5−458.i)T2 1 + (8.95 - 5.17i)T + (264.5 - 458. i)T^{2}
29 1+(30.4+17.5i)T+(420.5+728.i)T2 1 + (30.4 + 17.5i)T + (420.5 + 728. i)T^{2}
31 1+(−19.2−33.3i)T+(−480.5+832.i)T2 1 + (-19.2 - 33.3i)T + (-480.5 + 832. i)T^{2}
37 1−58.1T+1.36e3T2 1 - 58.1T + 1.36e3T^{2}
41 1+(11.9−6.92i)T+(840.5−1.45e3i)T2 1 + (11.9 - 6.92i)T + (840.5 - 1.45e3i)T^{2}
43 1+(8.84−15.3i)T+(−924.5−1.60e3i)T2 1 + (8.84 - 15.3i)T + (-924.5 - 1.60e3i)T^{2}
47 1+(−67.0−38.7i)T+(1.10e3+1.91e3i)T2 1 + (-67.0 - 38.7i)T + (1.10e3 + 1.91e3i)T^{2}
53 1+58.8iT−2.80e3T2 1 + 58.8iT - 2.80e3T^{2}
59 1+(35.0−20.2i)T+(1.74e3−3.01e3i)T2 1 + (35.0 - 20.2i)T + (1.74e3 - 3.01e3i)T^{2}
61 1+(27.1−46.9i)T+(−1.86e3−3.22e3i)T2 1 + (27.1 - 46.9i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−13.5−23.4i)T+(−2.24e3+3.88e3i)T2 1 + (-13.5 - 23.4i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1−1.77iT−5.04e3T2 1 - 1.77iT - 5.04e3T^{2}
73 1−128.T+5.32e3T2 1 - 128.T + 5.32e3T^{2}
79 1+(44.7−77.4i)T+(−3.12e3−5.40e3i)T2 1 + (44.7 - 77.4i)T + (-3.12e3 - 5.40e3i)T^{2}
83 1+(−15.7−9.08i)T+(3.44e3+5.96e3i)T2 1 + (-15.7 - 9.08i)T + (3.44e3 + 5.96e3i)T^{2}
89 1+84.2iT−7.92e3T2 1 + 84.2iT - 7.92e3T^{2}
97 1+(37.4−64.7i)T+(−4.70e3−8.14e3i)T2 1 + (37.4 - 64.7i)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

−13.32143904915307726174180028773, −12.28034837440069237123453022988, −11.10717290301032196370551660674, −9.864958407197055635418847604057, −9.401757656530165004181693778958, −7.25235059359507096690457312697, −6.08073553639815848529847940877, −5.18905957539448758885280226002, −4.25038674099498925759634655249, −1.11565100373678170315268059661, 2.13098114588776212177529118565, 4.05898372984993684490149530029, 5.49447388383585520910665762763, 6.35162291282058244088297414316, 7.88213024152732822798377741742, 9.279714830074225644896335472127, 10.59516373451177988568548604947, 11.55981987170713068059208098015, 12.32531150816771816516267553126, 13.29409316175257485271544741916

Graph of the ZZ-function along the critical line