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Properties

Label 2-637-7.2-c1-0-10
Degree 22
Conductor 637637
Sign −0.605+0.795i-0.605 + 0.795i
Analytic cond. 5.086475.08647
Root an. cond. 2.255322.25532
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  + (−0.933 + 1.61i)2-s + (1.67 + 2.90i)3-s + (−0.741 − 1.28i)4-s + (−0.433 + 0.750i)5-s − 6.24·6-s − 0.965·8-s + (−4.10 + 7.11i)9-s + (−0.808 − 1.39i)10-s + (1.93 + 3.34i)11-s + (2.48 − 4.30i)12-s + 13-s − 2.90·15-s + (2.38 − 4.12i)16-s + (1.67 + 2.90i)17-s + (−7.66 − 13.2i)18-s + (2.69 − 4.66i)19-s + ⋯
L(s)  = 1  + (−0.659 + 1.14i)2-s + (0.966 + 1.67i)3-s + (−0.370 − 0.642i)4-s + (−0.193 + 0.335i)5-s − 2.55·6-s − 0.341·8-s + (−1.36 + 2.37i)9-s + (−0.255 − 0.442i)10-s + (0.582 + 1.00i)11-s + (0.716 − 1.24i)12-s + 0.277·13-s − 0.748·15-s + (0.595 − 1.03i)16-s + (0.406 + 0.703i)17-s + (−1.80 − 3.12i)18-s + (0.617 − 1.06i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.605+0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.605+0.795i-0.605 + 0.795i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(79,⋅)\chi_{637} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.605+0.795i)(2,\ 637,\ (\ :1/2),\ -0.605 + 0.795i)

Particular Values

L(1)L(1) ≈\approx 0.593492−1.19721i0.593492 - 1.19721i
L(12)L(\frac12) ≈\approx 0.593492−1.19721i0.593492 - 1.19721i
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)
bad7 1 1
13 1−T 1 - T
good2 1+(0.933−1.61i)T+(−1−1.73i)T2 1 + (0.933 - 1.61i)T + (-1 - 1.73i)T^{2}
3 1+(−1.67−2.90i)T+(−1.5+2.59i)T2 1 + (-1.67 - 2.90i)T + (-1.5 + 2.59i)T^{2}
5 1+(0.433−0.750i)T+(−2.5−4.33i)T2 1 + (0.433 - 0.750i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1.93−3.34i)T+(−5.5+9.52i)T2 1 + (-1.93 - 3.34i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.67−2.90i)T+(−8.5+14.7i)T2 1 + (-1.67 - 2.90i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.69+4.66i)T+(−9.5−16.4i)T2 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.62+4.54i)T+(−11.5−19.9i)T2 1 + (-2.62 + 4.54i)T + (-11.5 - 19.9i)T^{2}
29 1−1.69T+29T2 1 - 1.69T + 29T^{2}
31 1+(3.78+6.55i)T+(−15.5+26.8i)T2 1 + (3.78 + 6.55i)T + (-15.5 + 26.8i)T^{2}
37 1+(−2.41+4.18i)T+(−18.5−32.0i)T2 1 + (-2.41 + 4.18i)T + (-18.5 - 32.0i)T^{2}
41 1+4.06T+41T2 1 + 4.06T + 41T^{2}
43 1−4.03T+43T2 1 - 4.03T + 43T^{2}
47 1+(−1.82+3.16i)T+(−23.5−40.7i)T2 1 + (-1.82 + 3.16i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.107−0.186i)T+(−26.5+45.8i)T2 1 + (-0.107 - 0.186i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1.39−2.41i)T+(−29.5+51.0i)T2 1 + (-1.39 - 2.41i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.51−7.82i)T+(−30.5−52.8i)T2 1 + (4.51 - 7.82i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.83−6.63i)T+(−33.5+58.0i)T2 1 + (-3.83 - 6.63i)T + (-33.5 + 58.0i)T^{2}
71 1−4.90T+71T2 1 - 4.90T + 71T^{2}
73 1+(7.77+13.4i)T+(−36.5+63.2i)T2 1 + (7.77 + 13.4i)T + (-36.5 + 63.2i)T^{2}
79 1+(4.71−8.16i)T+(−39.5−68.4i)T2 1 + (4.71 - 8.16i)T + (-39.5 - 68.4i)T^{2}
83 1+4.09T+83T2 1 + 4.09T + 83T^{2}
89 1+(−0.209+0.362i)T+(−44.5−77.0i)T2 1 + (-0.209 + 0.362i)T + (-44.5 - 77.0i)T^{2}
97 1+7.11T+97T2 1 + 7.11T + 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

−10.71835356600618835922273271249, −9.907553015569126528213349535960, −9.169121379783445561005655331658, −8.750255457015012957342084122698, −7.74278982051750506516985948995, −7.02777416599044518470485394884, −5.71582689338271742010158067868, −4.68944052630972722170433019945, −3.71405324026360066855495574597, −2.64590543205261566460566741738, 0.876548290787904101379704083952, 1.58387225089837139744746406110, 2.96550861823738999045185561556, 3.50262948687422162439013602052, 5.72534939361844554467738463343, 6.65225756346038314885482189277, 7.70445469248917101752760524612, 8.475884229959288557371536029933, 8.995400477218640636127098765262, 9.823320444566864810393589312746

Graph of the ZZ-function along the critical line