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Properties

Label 2-637-91.74-c1-0-23
Degree 22
Conductor 637637
Sign 0.312+0.949i0.312 + 0.949i
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  − 1.73·2-s + (0.366 + 0.633i)3-s + 0.999·4-s + (0.866 + 1.5i)5-s + (−0.633 − 1.09i)6-s + 1.73·8-s + (1.23 − 2.13i)9-s + (−1.49 − 2.59i)10-s + (−2.36 − 4.09i)11-s + (0.366 + 0.633i)12-s + (−1.59 − 3.23i)13-s + (−0.633 + 1.09i)15-s − 5·16-s − 4.26·17-s + (−2.13 + 3.69i)18-s + (−1 + 1.73i)19-s + ⋯
L(s)  = 1  − 1.22·2-s + (0.211 + 0.366i)3-s + 0.499·4-s + (0.387 + 0.670i)5-s + (−0.258 − 0.448i)6-s + 0.612·8-s + (0.410 − 0.711i)9-s + (−0.474 − 0.821i)10-s + (−0.713 − 1.23i)11-s + (0.105 + 0.183i)12-s + (−0.443 − 0.896i)13-s + (−0.163 + 0.283i)15-s − 1.25·16-s − 1.03·17-s + (−0.502 + 0.871i)18-s + (−0.229 + 0.397i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.312+0.949i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.312+0.949i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.312+0.949i0.312 + 0.949i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(165,⋅)\chi_{637} (165, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.312+0.949i)(2,\ 637,\ (\ :1/2),\ 0.312 + 0.949i)

Particular Values

L(1)L(1) ≈\approx 0.498628−0.360914i0.498628 - 0.360914i
L(12)L(\frac12) ≈\approx 0.498628−0.360914i0.498628 - 0.360914i
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+(1.59+3.23i)T 1 + (1.59 + 3.23i)T
good2 1+1.73T+2T2 1 + 1.73T + 2T^{2}
3 1+(−0.366−0.633i)T+(−1.5+2.59i)T2 1 + (-0.366 - 0.633i)T + (-1.5 + 2.59i)T^{2}
5 1+(−0.866−1.5i)T+(−2.5+4.33i)T2 1 + (-0.866 - 1.5i)T + (-2.5 + 4.33i)T^{2}
11 1+(2.36+4.09i)T+(−5.5+9.52i)T2 1 + (2.36 + 4.09i)T + (-5.5 + 9.52i)T^{2}
17 1+4.26T+17T2 1 + 4.26T + 17T^{2}
19 1+(1−1.73i)T+(−9.5−16.4i)T2 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2}
23 1−1.26T+23T2 1 - 1.26T + 23T^{2}
29 1+(−1.5+2.59i)T+(−14.5−25.1i)T2 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2}
31 1+(−3.09+5.36i)T+(−15.5−26.8i)T2 1 + (-3.09 + 5.36i)T + (-15.5 - 26.8i)T^{2}
37 1+7T+37T2 1 + 7T + 37T^{2}
41 1+(2.59−4.5i)T+(−20.5−35.5i)T2 1 + (2.59 - 4.5i)T + (-20.5 - 35.5i)T^{2}
43 1+(5.09+8.83i)T+(−21.5+37.2i)T2 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2}
47 1+(−0.464−0.803i)T+(−23.5+40.7i)T2 1 + (-0.464 - 0.803i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.96−3.40i)T+(−26.5−45.8i)T2 1 + (1.96 - 3.40i)T + (-26.5 - 45.8i)T^{2}
59 1−10.7T+59T2 1 - 10.7T + 59T^{2}
61 1+(−7.59+13.1i)T+(−30.5−52.8i)T2 1 + (-7.59 + 13.1i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.09+3.63i)T+(−33.5+58.0i)T2 1 + (2.09 + 3.63i)T + (-33.5 + 58.0i)T^{2}
71 1+(3+5.19i)T+(−35.5+61.4i)T2 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2}
73 1+(3.59−6.23i)T+(−36.5−63.2i)T2 1 + (3.59 - 6.23i)T + (-36.5 - 63.2i)T^{2}
79 1+(2.90+5.02i)T+(−39.5+68.4i)T2 1 + (2.90 + 5.02i)T + (-39.5 + 68.4i)T^{2}
83 1−8.19T+83T2 1 - 8.19T + 83T^{2}
89 1+0.928T+89T2 1 + 0.928T + 89T^{2}
97 1+(−7.19−12.4i)T+(−48.5+84.0i)T2 1 + (-7.19 - 12.4i)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

−10.33241210956498687273265791743, −9.626487628631171512667726836833, −8.658302454014191670095068495761, −8.120828848413968768641454745172, −7.01250330510461880530869810333, −6.16998439641410739154215555548, −4.86884107384667570159790280736, −3.51121697870780206351873417530, −2.34861426781548045296619249528, −0.49794417300932128086995355946, 1.52722091256529924963545745015, 2.31086174803166244226297971791, 4.58579823577039771669698339455, 4.98976548632169577198657356221, 6.97875970364840196587077416058, 7.17089685634967702469706586673, 8.452612441273263357510429085195, 8.856558367939913201908677850411, 9.868831522835656914984508154463, 10.34823426597866756210112819825

Graph of the ZZ-function along the critical line