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Properties

Label 2-930-31.5-c1-0-11
Degree 22
Conductor 930930
Sign 0.275+0.961i0.275 + 0.961i
Analytic cond. 7.426087.42608
Root an. cond. 2.725082.72508
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  − 2-s + (0.5 − 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (1.5 − 2.59i)7-s − 8-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (1.5 + 2.59i)11-s + (0.5 − 0.866i)12-s + (−2 − 3.46i)13-s + (−1.5 + 2.59i)14-s + 0.999·15-s + 16-s + (2 − 3.46i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s + (0.566 − 0.981i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.452 + 0.783i)11-s + (0.144 − 0.249i)12-s + (−0.554 − 0.960i)13-s + (−0.400 + 0.694i)14-s + 0.258·15-s + 0.250·16-s + (0.485 − 0.840i)17-s + ⋯

Functional equation

Λ(s)=(930s/2 ΓC(s) L(s)=((0.275+0.961i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(930s/2 ΓC(s+1/2) L(s)=((0.275+0.961i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 930930    =    2⋅3⋅5⋅312 \cdot 3 \cdot 5 \cdot 31
Sign: 0.275+0.961i0.275 + 0.961i
Analytic conductor: 7.426087.42608
Root analytic conductor: 2.725082.72508
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ930(811,⋅)\chi_{930} (811, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 930, ( :1/2), 0.275+0.961i)(2,\ 930,\ (\ :1/2),\ 0.275 + 0.961i)

Particular Values

L(1)L(1) ≈\approx 1.06765−0.805039i1.06765 - 0.805039i
L(12)L(\frac12) ≈\approx 1.06765−0.805039i1.06765 - 0.805039i
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)
bad2 1+T 1 + T
3 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
31 1+(3.5+4.33i)T 1 + (3.5 + 4.33i)T
good7 1+(−1.5+2.59i)T+(−3.5−6.06i)T2 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1.5−2.59i)T+(−5.5+9.52i)T2 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2}
13 1+(2+3.46i)T+(−6.5+11.2i)T2 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2}
17 1+(−2+3.46i)T+(−8.5−14.7i)T2 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2}
19 1+(−9.5−16.4i)T2 1 + (-9.5 - 16.4i)T^{2}
23 1−2T+23T2 1 - 2T + 23T^{2}
29 1−T+29T2 1 - T + 29T^{2}
37 1+(−3+5.19i)T+(−18.5−32.0i)T2 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1−1.73i)T+(−20.5+35.5i)T2 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2}
43 1+(2−3.46i)T+(−21.5−37.2i)T2 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2}
47 1−4T+47T2 1 - 4T + 47T^{2}
53 1+(−1.5−2.59i)T+(−26.5+45.8i)T2 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2}
59 1+(−4.5+7.79i)T+(−29.5−51.0i)T2 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+(1+1.73i)T+(−33.5+58.0i)T2 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2}
71 1+(2+3.46i)T+(−35.5+61.4i)T2 1 + (2 + 3.46i)T + (-35.5 + 61.4i)T^{2}
73 1+(1+1.73i)T+(−36.5+63.2i)T2 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2}
79 1+(−2+3.46i)T+(−39.5−68.4i)T2 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2}
83 1+(4.5+7.79i)T+(−41.5+71.8i)T2 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−11T+97T2 1 - 11T + 97T^{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

−9.830290058217555949582316951872, −9.187294919342815267317648382655, −7.949518320989708776429499444316, −7.47204458621880329207051667070, −6.89837681649799638410317250595, −5.72655549257289422337288064574, −4.53356909219669297757417948322, −3.22078953128042079407789685374, −2.09682530844649868937651155686, −0.824254013008525873639844661898, 1.49264767003436741142285984151, 2.60373812176604309632083487874, 3.88979642042421673933644973386, 5.08904831039211667915392591074, 5.87566347793163294273825009038, 6.92710271944468010358143007498, 8.106166451005508725014107280902, 8.742711826524714251020359689481, 9.187963647352689207280972916550, 10.07798911326150405737174883869

Graph of the ZZ-function along the critical line