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Properties

Label 2-637-13.9-c1-0-11
Degree 22
Conductor 637637
Sign 0.384−0.922i0.384 - 0.922i
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  + (0.760 − 1.31i)2-s + (−1.06 + 1.84i)3-s + (−0.156 − 0.270i)4-s − 0.589·5-s + (1.61 + 2.80i)6-s + 2.56·8-s + (−0.760 − 1.31i)9-s + (−0.448 + 0.776i)10-s + (−0.760 + 1.31i)11-s + 0.664·12-s + (−3.32 + 1.39i)13-s + (0.626 − 1.08i)15-s + (2.26 − 3.92i)16-s + (2.39 + 4.15i)17-s − 2.31·18-s + (0.841 + 1.45i)19-s + ⋯
L(s)  = 1  + (0.537 − 0.931i)2-s + (−0.613 + 1.06i)3-s + (−0.0781 − 0.135i)4-s − 0.263·5-s + (0.660 + 1.14i)6-s + 0.907·8-s + (−0.253 − 0.439i)9-s + (−0.141 + 0.245i)10-s + (−0.229 + 0.397i)11-s + 0.191·12-s + (−0.922 + 0.386i)13-s + (0.161 − 0.280i)15-s + (0.565 − 0.980i)16-s + (0.581 + 1.00i)17-s − 0.545·18-s + (0.193 + 0.334i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.384−0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.384−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.384−0.922i0.384 - 0.922i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(295,⋅)\chi_{637} (295, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.384−0.922i)(2,\ 637,\ (\ :1/2),\ 0.384 - 0.922i)

Particular Values

L(1)L(1) ≈\approx 1.19100+0.793666i1.19100 + 0.793666i
L(12)L(\frac12) ≈\approx 1.19100+0.793666i1.19100 + 0.793666i
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+(3.32−1.39i)T 1 + (3.32 - 1.39i)T
good2 1+(−0.760+1.31i)T+(−1−1.73i)T2 1 + (-0.760 + 1.31i)T + (-1 - 1.73i)T^{2}
3 1+(1.06−1.84i)T+(−1.5−2.59i)T2 1 + (1.06 - 1.84i)T + (-1.5 - 2.59i)T^{2}
5 1+0.589T+5T2 1 + 0.589T + 5T^{2}
11 1+(0.760−1.31i)T+(−5.5−9.52i)T2 1 + (0.760 - 1.31i)T + (-5.5 - 9.52i)T^{2}
17 1+(−2.39−4.15i)T+(−8.5+14.7i)T2 1 + (-2.39 - 4.15i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.841−1.45i)T+(−9.5+16.4i)T2 1 + (-0.841 - 1.45i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.886−1.53i)T+(−11.5−19.9i)T2 1 + (0.886 - 1.53i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.44−5.96i)T+(−14.5−25.1i)T2 1 + (3.44 - 5.96i)T + (-14.5 - 25.1i)T^{2}
31 1−6.08T+31T2 1 - 6.08T + 31T^{2}
37 1+(0.704−1.22i)T+(−18.5−32.0i)T2 1 + (0.704 - 1.22i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.677+1.17i)T+(−20.5−35.5i)T2 1 + (-0.677 + 1.17i)T + (-20.5 - 35.5i)T^{2}
43 1+(−5.77−10.0i)T+(−21.5+37.2i)T2 1 + (-5.77 - 10.0i)T + (-21.5 + 37.2i)T^{2}
47 1−0.464T+47T2 1 - 0.464T + 47T^{2}
53 1−8.24T+53T2 1 - 8.24T + 53T^{2}
59 1+(5.93+10.2i)T+(−29.5+51.0i)T2 1 + (5.93 + 10.2i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.24−2.14i)T+(−30.5+52.8i)T2 1 + (-1.24 - 2.14i)T + (-30.5 + 52.8i)T^{2}
67 1+(−3.78+6.55i)T+(−33.5−58.0i)T2 1 + (-3.78 + 6.55i)T + (-33.5 - 58.0i)T^{2}
71 1+(3.30+5.71i)T+(−35.5+61.4i)T2 1 + (3.30 + 5.71i)T + (-35.5 + 61.4i)T^{2}
73 1−16.3T+73T2 1 - 16.3T + 73T^{2}
79 1+14.9T+79T2 1 + 14.9T + 79T^{2}
83 1+10.1T+83T2 1 + 10.1T + 83T^{2}
89 1+(−8.24+14.2i)T+(−44.5−77.0i)T2 1 + (-8.24 + 14.2i)T + (-44.5 - 77.0i)T^{2}
97 1+(0.486+0.843i)T+(−48.5+84.0i)T2 1 + (0.486 + 0.843i)T + (-48.5 + 84.0i)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

−10.83792271760549351377683655653, −10.07530471665828623844191563603, −9.590834576514112787267966791358, −8.072086312271248416643421007843, −7.28766277332740391763409173554, −5.83117510110445585152383349944, −4.85252188583718203242907420173, −4.18959726849041110709906162794, −3.29057157443323716982563814079, −1.87272041631664334366659867596, 0.71053457165757075059635209742, 2.40507548721078497229187371039, 4.14251011538571415330961171129, 5.36054392558397897088730336285, 5.85667906557422222491843185524, 6.90745429644619070007091050113, 7.44585344208448070796575038553, 8.098082563735665238727703414560, 9.586035055152074136869054940301, 10.49876496323365772648126235188

Graph of the ZZ-function along the critical line