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Properties

Label 2-117-117.41-c1-0-4
Degree 22
Conductor 117117
Sign 0.382−0.923i0.382 - 0.923i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0826 − 0.0221i)2-s + (0.405 + 1.68i)3-s + (−1.72 + 0.996i)4-s + (1.73 − 0.466i)5-s + (0.0707 + 0.130i)6-s + (0.362 + 0.362i)7-s + (−0.241 + 0.241i)8-s + (−2.67 + 1.36i)9-s + (0.133 − 0.0770i)10-s + (1.48 + 5.53i)11-s + (−2.37 − 2.50i)12-s + (−0.119 − 3.60i)13-s + (0.0379 + 0.0219i)14-s + (1.48 + 2.74i)15-s + (1.97 − 3.42i)16-s + (2.73 − 4.74i)17-s + ⋯
L(s)  = 1  + (0.0584 − 0.0156i)2-s + (0.233 + 0.972i)3-s + (−0.862 + 0.498i)4-s + (0.777 − 0.208i)5-s + (0.0288 + 0.0531i)6-s + (0.136 + 0.136i)7-s + (−0.0853 + 0.0853i)8-s + (−0.890 + 0.454i)9-s + (0.0421 − 0.0243i)10-s + (0.447 + 1.66i)11-s + (−0.686 − 0.722i)12-s + (−0.0332 − 0.999i)13-s + (0.0101 + 0.00585i)14-s + (0.384 + 0.707i)15-s + (0.494 − 0.856i)16-s + (0.663 − 1.14i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.382−0.923i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.382−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.382−0.923i0.382 - 0.923i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(41,⋅)\chi_{117} (41, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.382−0.923i)(2,\ 117,\ (\ :1/2),\ 0.382 - 0.923i)

Particular Values

L(1)L(1) ≈\approx 0.899619+0.601013i0.899619 + 0.601013i
L(12)L(\frac12) ≈\approx 0.899619+0.601013i0.899619 + 0.601013i
L(32)L(\frac{3}{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+(−0.405−1.68i)T 1 + (-0.405 - 1.68i)T
13 1+(0.119+3.60i)T 1 + (0.119 + 3.60i)T
good2 1+(−0.0826+0.0221i)T+(1.73−i)T2 1 + (-0.0826 + 0.0221i)T + (1.73 - i)T^{2}
5 1+(−1.73+0.466i)T+(4.33−2.5i)T2 1 + (-1.73 + 0.466i)T + (4.33 - 2.5i)T^{2}
7 1+(−0.362−0.362i)T+7iT2 1 + (-0.362 - 0.362i)T + 7iT^{2}
11 1+(−1.48−5.53i)T+(−9.52+5.5i)T2 1 + (-1.48 - 5.53i)T + (-9.52 + 5.5i)T^{2}
17 1+(−2.73+4.74i)T+(−8.5−14.7i)T2 1 + (-2.73 + 4.74i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.92+7.17i)T+(−16.4+9.5i)T2 1 + (1.92 + 7.17i)T + (-16.4 + 9.5i)T^{2}
23 1−4.43T+23T2 1 - 4.43T + 23T^{2}
29 1+(−0.446−0.257i)T+(14.5+25.1i)T2 1 + (-0.446 - 0.257i)T + (14.5 + 25.1i)T^{2}
31 1+(−0.854−3.18i)T+(−26.8+15.5i)T2 1 + (-0.854 - 3.18i)T + (-26.8 + 15.5i)T^{2}
37 1+(0.815−3.04i)T+(−32.0−18.5i)T2 1 + (0.815 - 3.04i)T + (-32.0 - 18.5i)T^{2}
41 1+(1.97+1.97i)T+41iT2 1 + (1.97 + 1.97i)T + 41iT^{2}
43 1−2.09iT−43T2 1 - 2.09iT - 43T^{2}
47 1+(4.06+1.08i)T+(40.7+23.5i)T2 1 + (4.06 + 1.08i)T + (40.7 + 23.5i)T^{2}
53 1−8.79iT−53T2 1 - 8.79iT - 53T^{2}
59 1+(−7.47−2.00i)T+(51.0+29.5i)T2 1 + (-7.47 - 2.00i)T + (51.0 + 29.5i)T^{2}
61 1+5.79T+61T2 1 + 5.79T + 61T^{2}
67 1+(3.78−3.78i)T−67iT2 1 + (3.78 - 3.78i)T - 67iT^{2}
71 1+(2.49−0.668i)T+(61.4−35.5i)T2 1 + (2.49 - 0.668i)T + (61.4 - 35.5i)T^{2}
73 1+(4.21+4.21i)T+73iT2 1 + (4.21 + 4.21i)T + 73iT^{2}
79 1+(2.43+4.20i)T+(−39.5+68.4i)T2 1 + (2.43 + 4.20i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.879−3.28i)T+(−71.8−41.5i)T2 1 + (0.879 - 3.28i)T + (-71.8 - 41.5i)T^{2}
89 1+(10.8+2.90i)T+(77.0+44.5i)T2 1 + (10.8 + 2.90i)T + (77.0 + 44.5i)T^{2}
97 1+(1.56−1.56i)T−97iT2 1 + (1.56 - 1.56i)T - 97iT^{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

−13.70812977730793009421331694727, −12.88881226204245830541152717905, −11.73175082916765137708568108868, −10.20652234201906845179385271069, −9.488217687072945090883411026209, −8.754218936093945266443235422101, −7.30481933082032617143508969696, −5.25513389688950504716651635709, −4.61417131389974173991234271356, −2.91010402478407962891029983903, 1.51597348080417927992314190880, 3.70218679983023759125284803709, 5.77158690682178030871000058272, 6.31608307121196184058266203225, 8.124195627414708169028494960772, 8.918161076508812338040249609590, 10.09667248374199745858143004971, 11.31342988667296218104894409960, 12.60501789449499822439418377412, 13.54809398293390575699594912117

Graph of the ZZ-function along the critical line