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Properties

Label 2-117-13.7-c2-0-7
Degree 22
Conductor 117117
Sign 0.964+0.264i0.964 + 0.264i
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  + (3.06 − 0.822i)2-s + (5.27 − 3.04i)4-s + (5.58 + 5.58i)5-s + (−7.30 − 1.95i)7-s + (4.69 − 4.69i)8-s + (21.7 + 12.5i)10-s + (−3.01 − 11.2i)11-s + (−12.4 − 3.76i)13-s − 24.0·14-s + (−1.64 + 2.84i)16-s + (10.4 − 6.05i)17-s + (2.73 − 10.1i)19-s + (46.4 + 12.4i)20-s + (−18.4 − 32.0i)22-s + (3.75 + 2.16i)23-s + ⋯
L(s)  = 1  + (1.53 − 0.411i)2-s + (1.31 − 0.761i)4-s + (1.11 + 1.11i)5-s + (−1.04 − 0.279i)7-s + (0.586 − 0.586i)8-s + (2.17 + 1.25i)10-s + (−0.273 − 1.02i)11-s + (−0.957 − 0.289i)13-s − 1.71·14-s + (−0.102 + 0.177i)16-s + (0.616 − 0.355i)17-s + (0.143 − 0.536i)19-s + (2.32 + 0.622i)20-s + (−0.840 − 1.45i)22-s + (0.163 + 0.0942i)23-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.964+0.264i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((0.964+0.264i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.964+0.264i0.964 + 0.264i
Analytic conductor: 3.188013.18801
Root analytic conductor: 1.785501.78550
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ117(46,⋅)\chi_{117} (46, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1), 0.964+0.264i)(2,\ 117,\ (\ :1),\ 0.964 + 0.264i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.98845−0.402942i2.98845 - 0.402942i
L(12)L(\frac12) ≈\approx 2.98845−0.402942i2.98845 - 0.402942i
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+(12.4+3.76i)T 1 + (12.4 + 3.76i)T
good2 1+(−3.06+0.822i)T+(3.46−2i)T2 1 + (-3.06 + 0.822i)T + (3.46 - 2i)T^{2}
5 1+(−5.58−5.58i)T+25iT2 1 + (-5.58 - 5.58i)T + 25iT^{2}
7 1+(7.30+1.95i)T+(42.4+24.5i)T2 1 + (7.30 + 1.95i)T + (42.4 + 24.5i)T^{2}
11 1+(3.01+11.2i)T+(−104.+60.5i)T2 1 + (3.01 + 11.2i)T + (-104. + 60.5i)T^{2}
17 1+(−10.4+6.05i)T+(144.5−250.i)T2 1 + (-10.4 + 6.05i)T + (144.5 - 250. i)T^{2}
19 1+(−2.73+10.1i)T+(−312.−180.5i)T2 1 + (-2.73 + 10.1i)T + (-312. - 180.5i)T^{2}
23 1+(−3.75−2.16i)T+(264.5+458.i)T2 1 + (-3.75 - 2.16i)T + (264.5 + 458. i)T^{2}
29 1+(22.1−38.3i)T+(−420.5−728.i)T2 1 + (22.1 - 38.3i)T + (-420.5 - 728. i)T^{2}
31 1+(5.71+5.71i)T+961iT2 1 + (5.71 + 5.71i)T + 961iT^{2}
37 1+(−8.51−31.7i)T+(−1.18e3+684.5i)T2 1 + (-8.51 - 31.7i)T + (-1.18e3 + 684.5i)T^{2}
41 1+(−58.1+15.5i)T+(1.45e3−840.5i)T2 1 + (-58.1 + 15.5i)T + (1.45e3 - 840.5i)T^{2}
43 1+(−27.0+15.5i)T+(924.5−1.60e3i)T2 1 + (-27.0 + 15.5i)T + (924.5 - 1.60e3i)T^{2}
47 1+(22.4−22.4i)T−2.20e3iT2 1 + (22.4 - 22.4i)T - 2.20e3iT^{2}
53 1−54.7T+2.80e3T2 1 - 54.7T + 2.80e3T^{2}
59 1+(−13.4−3.61i)T+(3.01e3+1.74e3i)T2 1 + (-13.4 - 3.61i)T + (3.01e3 + 1.74e3i)T^{2}
61 1+(42.0+72.8i)T+(−1.86e3+3.22e3i)T2 1 + (42.0 + 72.8i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(−62.2+16.6i)T+(3.88e3−2.24e3i)T2 1 + (-62.2 + 16.6i)T + (3.88e3 - 2.24e3i)T^{2}
71 1+(−14.8+55.2i)T+(−4.36e3−2.52e3i)T2 1 + (-14.8 + 55.2i)T + (-4.36e3 - 2.52e3i)T^{2}
73 1+(54.9−54.9i)T−5.32e3iT2 1 + (54.9 - 54.9i)T - 5.32e3iT^{2}
79 1+45.0T+6.24e3T2 1 + 45.0T + 6.24e3T^{2}
83 1+(−30.8−30.8i)T+6.88e3iT2 1 + (-30.8 - 30.8i)T + 6.88e3iT^{2}
89 1+(5.84+21.8i)T+(−6.85e3+3.96e3i)T2 1 + (5.84 + 21.8i)T + (-6.85e3 + 3.96e3i)T^{2}
97 1+(−28.7+107.i)T+(−8.14e3−4.70e3i)T2 1 + (-28.7 + 107. i)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

−13.32320845035805097652694742830, −12.59868268691812867175753678674, −11.24106084971998889082682210719, −10.43000430281335636497624081976, −9.417201797359041677340228281251, −7.15773741436931077895895141280, −6.16485215216766736646164623872, −5.29960695127949138926461678966, −3.37312451787555492692349172075, −2.63424611563544666976536831001, 2.39878544364948873692355449719, 4.22710528124670227309935638839, 5.38019209858196876565615055410, 6.10823049673208701997054505840, 7.44618534116947661133842645631, 9.334018822123375804783373238187, 9.931503867446823317085929500969, 12.00966329077339014054747601190, 12.75102435290159903092945112886, 13.10217236651893767145714461385

Graph of the ZZ-function along the critical line