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Properties

Label 2-117-117.59-c1-0-7
Degree 22
Conductor 117117
Sign 0.984−0.173i0.984 - 0.173i
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  + (0.311 − 0.311i)2-s + (1.53 + 0.809i)3-s + 1.80i·4-s + (−0.789 − 2.94i)5-s + (0.729 − 0.224i)6-s + (0.0751 + 0.280i)7-s + (1.18 + 1.18i)8-s + (1.68 + 2.47i)9-s + (−1.16 − 0.671i)10-s + (−1.04 − 1.04i)11-s + (−1.46 + 2.76i)12-s + (−3.58 − 0.362i)13-s + (0.110 + 0.0639i)14-s + (1.17 − 5.14i)15-s − 2.87·16-s + (0.767 + 1.32i)17-s + ⋯
L(s)  = 1  + (0.220 − 0.220i)2-s + (0.884 + 0.467i)3-s + 0.902i·4-s + (−0.352 − 1.31i)5-s + (0.297 − 0.0917i)6-s + (0.0284 + 0.106i)7-s + (0.419 + 0.419i)8-s + (0.563 + 0.826i)9-s + (−0.367 − 0.212i)10-s + (−0.316 − 0.316i)11-s + (−0.422 + 0.798i)12-s + (−0.994 − 0.100i)13-s + (0.0296 + 0.0170i)14-s + (0.303 − 1.32i)15-s − 0.718·16-s + (0.186 + 0.322i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.984−0.173i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.984−0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.984−0.173i0.984 - 0.173i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(59,⋅)\chi_{117} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.984−0.173i)(2,\ 117,\ (\ :1/2),\ 0.984 - 0.173i)

Particular Values

L(1)L(1) ≈\approx 1.38054+0.120541i1.38054 + 0.120541i
L(12)L(\frac12) ≈\approx 1.38054+0.120541i1.38054 + 0.120541i
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+(−1.53−0.809i)T 1 + (-1.53 - 0.809i)T
13 1+(3.58+0.362i)T 1 + (3.58 + 0.362i)T
good2 1+(−0.311+0.311i)T−2iT2 1 + (-0.311 + 0.311i)T - 2iT^{2}
5 1+(0.789+2.94i)T+(−4.33+2.5i)T2 1 + (0.789 + 2.94i)T + (-4.33 + 2.5i)T^{2}
7 1+(−0.0751−0.280i)T+(−6.06+3.5i)T2 1 + (-0.0751 - 0.280i)T + (-6.06 + 3.5i)T^{2}
11 1+(1.04+1.04i)T+11iT2 1 + (1.04 + 1.04i)T + 11iT^{2}
17 1+(−0.767−1.32i)T+(−8.5+14.7i)T2 1 + (-0.767 - 1.32i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.989+3.69i)T+(−16.4−9.5i)T2 1 + (-0.989 + 3.69i)T + (-16.4 - 9.5i)T^{2}
23 1+(3.92+6.79i)T+(−11.5+19.9i)T2 1 + (3.92 + 6.79i)T + (-11.5 + 19.9i)T^{2}
29 1−1.80iT−29T2 1 - 1.80iT - 29T^{2}
31 1+(3.94−1.05i)T+(26.8−15.5i)T2 1 + (3.94 - 1.05i)T + (26.8 - 15.5i)T^{2}
37 1+(−2.40−8.97i)T+(−32.0+18.5i)T2 1 + (-2.40 - 8.97i)T + (-32.0 + 18.5i)T^{2}
41 1+(2.37+0.636i)T+(35.5+20.5i)T2 1 + (2.37 + 0.636i)T + (35.5 + 20.5i)T^{2}
43 1+(−8.95−5.16i)T+(21.5+37.2i)T2 1 + (-8.95 - 5.16i)T + (21.5 + 37.2i)T^{2}
47 1+(−2.30+8.58i)T+(−40.7−23.5i)T2 1 + (-2.30 + 8.58i)T + (-40.7 - 23.5i)T^{2}
53 1−8.42iT−53T2 1 - 8.42iT - 53T^{2}
59 1+(−8.77−8.77i)T+59iT2 1 + (-8.77 - 8.77i)T + 59iT^{2}
61 1+(−3.29+5.71i)T+(−30.5−52.8i)T2 1 + (-3.29 + 5.71i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.95−7.28i)T+(−58.0−33.5i)T2 1 + (1.95 - 7.28i)T + (-58.0 - 33.5i)T^{2}
71 1+(4.14+1.11i)T+(61.4+35.5i)T2 1 + (4.14 + 1.11i)T + (61.4 + 35.5i)T^{2}
73 1+(−4.56+4.56i)T−73iT2 1 + (-4.56 + 4.56i)T - 73iT^{2}
79 1+(2.96+5.13i)T+(−39.5+68.4i)T2 1 + (2.96 + 5.13i)T + (-39.5 + 68.4i)T^{2}
83 1+(11.4+3.07i)T+(71.8+41.5i)T2 1 + (11.4 + 3.07i)T + (71.8 + 41.5i)T^{2}
89 1+(1.37−0.367i)T+(77.0−44.5i)T2 1 + (1.37 - 0.367i)T + (77.0 - 44.5i)T^{2}
97 1+(−3.26+0.874i)T+(84.0−48.5i)T2 1 + (-3.26 + 0.874i)T + (84.0 - 48.5i)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

−13.33675629931404492521413020628, −12.66696187895202277521345313235, −11.80932627410440907029727425533, −10.34065304502800930308860479322, −8.975878119276239338249095124219, −8.384837530354157857301099400165, −7.41929409445412038842931547570, −5.00784692986418132793111549501, −4.14556669020168338689504649882, −2.62167191824847110212356302328, 2.26772852025973535844186357646, 3.87118651219387596835552445318, 5.76088233996956687008911564942, 7.14323387763798027139412667816, 7.60704886589443642612733730083, 9.482086974319521463507012763948, 10.18971644698547945049570712524, 11.39291991324261792104977460819, 12.66171930373513034237706036805, 14.03414228539307212385823320099

Graph of the ZZ-function along the critical line