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Properties

Label 2-637-13.9-c1-0-41
Degree 22
Conductor 637637
Sign −0.617−0.786i-0.617 - 0.786i
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  + (1.16 − 2.01i)2-s + (1.15 − 1.99i)3-s + (−1.71 − 2.97i)4-s − 3.37·5-s + (−2.69 − 4.66i)6-s − 3.34·8-s + (−1.16 − 2.01i)9-s + (−3.92 + 6.80i)10-s + (−1.16 + 2.01i)11-s − 7.93·12-s + (−0.408 + 3.58i)13-s + (−3.89 + 6.74i)15-s + (−0.466 + 0.808i)16-s + (−2.72 − 4.72i)17-s − 5.43·18-s + (−3.58 − 6.20i)19-s + ⋯
L(s)  = 1  + (0.824 − 1.42i)2-s + (0.666 − 1.15i)3-s + (−0.858 − 1.48i)4-s − 1.50·5-s + (−1.09 − 1.90i)6-s − 1.18·8-s + (−0.388 − 0.673i)9-s + (−1.24 + 2.15i)10-s + (−0.351 + 0.608i)11-s − 2.29·12-s + (−0.113 + 0.993i)13-s + (−1.00 + 1.74i)15-s + (−0.116 + 0.202i)16-s + (−0.661 − 1.14i)17-s − 1.28·18-s + (−0.822 − 1.42i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.617−0.786i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.617−0.786i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.617−0.786i-0.617 - 0.786i
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), −0.617−0.786i)(2,\ 637,\ (\ :1/2),\ -0.617 - 0.786i)

Particular Values

L(1)L(1) ≈\approx 0.763004+1.56906i0.763004 + 1.56906i
L(12)L(\frac12) ≈\approx 0.763004+1.56906i0.763004 + 1.56906i
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+(0.408−3.58i)T 1 + (0.408 - 3.58i)T
good2 1+(−1.16+2.01i)T+(−1−1.73i)T2 1 + (-1.16 + 2.01i)T + (-1 - 1.73i)T^{2}
3 1+(−1.15+1.99i)T+(−1.5−2.59i)T2 1 + (-1.15 + 1.99i)T + (-1.5 - 2.59i)T^{2}
5 1+3.37T+5T2 1 + 3.37T + 5T^{2}
11 1+(1.16−2.01i)T+(−5.5−9.52i)T2 1 + (1.16 - 2.01i)T + (-5.5 - 9.52i)T^{2}
17 1+(2.72+4.72i)T+(−8.5+14.7i)T2 1 + (2.72 + 4.72i)T + (-8.5 + 14.7i)T^{2}
19 1+(3.58+6.20i)T+(−9.5+16.4i)T2 1 + (3.58 + 6.20i)T + (-9.5 + 16.4i)T^{2}
23 1+(−3.22+5.58i)T+(−11.5−19.9i)T2 1 + (-3.22 + 5.58i)T + (-11.5 - 19.9i)T^{2}
29 1+(−4.22+7.31i)T+(−14.5−25.1i)T2 1 + (-4.22 + 7.31i)T + (-14.5 - 25.1i)T^{2}
31 1−3.05T+31T2 1 - 3.05T + 31T^{2}
37 1+(1.52−2.64i)T+(−18.5−32.0i)T2 1 + (1.52 - 2.64i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.468−0.812i)T+(−20.5−35.5i)T2 1 + (0.468 - 0.812i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.04−3.54i)T+(−21.5+37.2i)T2 1 + (-2.04 - 3.54i)T + (-21.5 + 37.2i)T^{2}
47 1−3.46T+47T2 1 - 3.46T + 47T^{2}
53 1+2.34T+53T2 1 + 2.34T + 53T^{2}
59 1+(−3.62−6.27i)T+(−29.5+51.0i)T2 1 + (-3.62 - 6.27i)T + (-29.5 + 51.0i)T^{2}
61 1+(−3.19−5.53i)T+(−30.5+52.8i)T2 1 + (-3.19 - 5.53i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.30−3.99i)T+(−33.5−58.0i)T2 1 + (2.30 - 3.99i)T + (-33.5 - 58.0i)T^{2}
71 1+(−3.79−6.57i)T+(−35.5+61.4i)T2 1 + (-3.79 - 6.57i)T + (-35.5 + 61.4i)T^{2}
73 1−2.06T+73T2 1 - 2.06T + 73T^{2}
79 1+7.58T+79T2 1 + 7.58T + 79T^{2}
83 1−2.89T+83T2 1 - 2.89T + 83T^{2}
89 1+(−6.57+11.3i)T+(−44.5−77.0i)T2 1 + (-6.57 + 11.3i)T + (-44.5 - 77.0i)T^{2}
97 1+(1.77+3.08i)T+(−48.5+84.0i)T2 1 + (1.77 + 3.08i)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

−10.37334963675115886883640711211, −9.131833836278749519188007707735, −8.323103837724361464183753096144, −7.27595283718732150154225342631, −6.71952732512075635881454491976, −4.56515465127455670268489981780, −4.44224781987785919478104501958, −2.82971545980281424348696263257, −2.34584168565053353127241613396, −0.68030151007276984111322096984, 3.38480474173710831417303755515, 3.73248049496366691386196737027, 4.64399925509776697589490132748, 5.54105787324750404658134547920, 6.68238508044801408808555433789, 7.82592208360553833677156465455, 8.238328857462553614457021615018, 8.891813141182626132236092556628, 10.39303988952864120590059028332, 10.90589391464520607817913012072

Graph of the ZZ-function along the critical line