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Properties

Label 2-117-39.29-c2-0-0
Degree 22
Conductor 117117
Sign −0.999−0.0236i-0.999 - 0.0236i
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  + (−1.02 + 0.592i)2-s + (−1.29 + 2.24i)4-s − 4.21i·5-s + (−4.97 + 8.62i)7-s − 7.81i·8-s + (2.49 + 4.32i)10-s + (−16.4 + 9.48i)11-s + (−4.13 − 12.3i)13-s − 11.7i·14-s + (−0.568 − 0.985i)16-s + (−7.09 − 4.09i)17-s + (−11.0 + 19.1i)19-s + (9.48 + 5.47i)20-s + (11.2 − 19.4i)22-s + (−17.6 + 10.1i)23-s + ⋯
L(s)  = 1  + (−0.512 + 0.296i)2-s + (−0.324 + 0.562i)4-s − 0.843i·5-s + (−0.711 + 1.23i)7-s − 0.976i·8-s + (0.249 + 0.432i)10-s + (−1.49 + 0.862i)11-s + (−0.317 − 0.948i)13-s − 0.842i·14-s + (−0.0355 − 0.0615i)16-s + (−0.417 − 0.241i)17-s + (−0.582 + 1.00i)19-s + (0.474 + 0.273i)20-s + (0.510 − 0.884i)22-s + (−0.765 + 0.442i)23-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((−0.999−0.0236i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0236i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1) L(s)=((−0.999−0.0236i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0236i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: −0.999−0.0236i-0.999 - 0.0236i
Analytic conductor: 3.188013.18801
Root analytic conductor: 1.785501.78550
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ117(107,⋅)\chi_{117} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1), −0.999−0.0236i)(2,\ 117,\ (\ :1),\ -0.999 - 0.0236i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.00322433+0.272381i0.00322433 + 0.272381i
L(12)L(\frac12) ≈\approx 0.00322433+0.272381i0.00322433 + 0.272381i
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+(4.13+12.3i)T 1 + (4.13 + 12.3i)T
good2 1+(1.02−0.592i)T+(2−3.46i)T2 1 + (1.02 - 0.592i)T + (2 - 3.46i)T^{2}
5 1+4.21iT−25T2 1 + 4.21iT - 25T^{2}
7 1+(4.97−8.62i)T+(−24.5−42.4i)T2 1 + (4.97 - 8.62i)T + (-24.5 - 42.4i)T^{2}
11 1+(16.4−9.48i)T+(60.5−104.i)T2 1 + (16.4 - 9.48i)T + (60.5 - 104. i)T^{2}
17 1+(7.09+4.09i)T+(144.5+250.i)T2 1 + (7.09 + 4.09i)T + (144.5 + 250. i)T^{2}
19 1+(11.0−19.1i)T+(−180.5−312.i)T2 1 + (11.0 - 19.1i)T + (-180.5 - 312. i)T^{2}
23 1+(17.6−10.1i)T+(264.5−458.i)T2 1 + (17.6 - 10.1i)T + (264.5 - 458. i)T^{2}
29 1+(−9.66+5.57i)T+(420.5−728.i)T2 1 + (-9.66 + 5.57i)T + (420.5 - 728. i)T^{2}
31 1−11.3T+961T2 1 - 11.3T + 961T^{2}
37 1+(−25.2−43.6i)T+(−684.5+1.18e3i)T2 1 + (-25.2 - 43.6i)T + (-684.5 + 1.18e3i)T^{2}
41 1+(−9.23+5.33i)T+(840.5−1.45e3i)T2 1 + (-9.23 + 5.33i)T + (840.5 - 1.45e3i)T^{2}
43 1+(−30.1+52.1i)T+(−924.5−1.60e3i)T2 1 + (-30.1 + 52.1i)T + (-924.5 - 1.60e3i)T^{2}
47 1−71.8iT−2.20e3T2 1 - 71.8iT - 2.20e3T^{2}
53 1−12.1iT−2.80e3T2 1 - 12.1iT - 2.80e3T^{2}
59 1+(43.4+25.1i)T+(1.74e3+3.01e3i)T2 1 + (43.4 + 25.1i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(31.6−54.8i)T+(−1.86e3−3.22e3i)T2 1 + (31.6 - 54.8i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(36.7+63.6i)T+(−2.24e3+3.88e3i)T2 1 + (36.7 + 63.6i)T + (-2.24e3 + 3.88e3i)T^{2}
71 1+(84.7+48.9i)T+(2.52e3+4.36e3i)T2 1 + (84.7 + 48.9i)T + (2.52e3 + 4.36e3i)T^{2}
73 1−9.79T+5.32e3T2 1 - 9.79T + 5.32e3T^{2}
79 1+17.6T+6.24e3T2 1 + 17.6T + 6.24e3T^{2}
83 1−13.4iT−6.88e3T2 1 - 13.4iT - 6.88e3T^{2}
89 1+(143.−83.0i)T+(3.96e3−6.85e3i)T2 1 + (143. - 83.0i)T + (3.96e3 - 6.85e3i)T^{2}
97 1+(53.9−93.4i)T+(−4.70e3−8.14e3i)T2 1 + (53.9 - 93.4i)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

−13.43604911743916067964168482849, −12.48209122367487680196326069182, −12.37219072659551869683126469865, −10.27433509618899227676565235623, −9.416609328141303647486785419469, −8.402596863685913262899236293884, −7.63880632841028694239039277822, −5.93416614306719412719490583954, −4.67555486449492630533226253282, −2.76338745102619428089748818268, 0.21646368705124327691493199509, 2.64966639719665695037307089638, 4.45353632572984827610588953786, 6.13560181458890394945539965163, 7.25082428153658241872637847467, 8.605003574319413291860313141310, 9.883044009218133080443989390253, 10.63652151737667072518578390215, 11.16578591122558418034938377567, 13.03680440870399575162926685041

Graph of the ZZ-function along the critical line