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Properties

Label 2-637-13.3-c1-0-8
Degree 22
Conductor 637637
Sign −0.929−0.367i-0.929 - 0.367i
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  + (0.651 + 1.12i)2-s + (1.44 + 2.49i)3-s + (0.151 − 0.262i)4-s − 2.88·5-s + (−1.87 + 3.25i)6-s + 3·8-s + (−2.65 + 4.59i)9-s + (−1.87 − 3.25i)10-s + (2.95 + 5.11i)11-s + 0.872·12-s + (−3.31 + 1.41i)13-s + (−4.15 − 7.19i)15-s + (1.65 + 2.86i)16-s + (−0.436 + 0.755i)17-s − 6.90·18-s + (1.44 − 2.49i)19-s + ⋯
L(s)  = 1  + (0.460 + 0.797i)2-s + (0.831 + 1.44i)3-s + (0.0756 − 0.131i)4-s − 1.28·5-s + (−0.766 + 1.32i)6-s + 1.06·8-s + (−0.883 + 1.53i)9-s + (−0.593 − 1.02i)10-s + (0.890 + 1.54i)11-s + 0.251·12-s + (−0.920 + 0.391i)13-s + (−1.07 − 1.85i)15-s + (0.412 + 0.715i)16-s + (−0.105 + 0.183i)17-s − 1.62·18-s + (0.330 − 0.572i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.929−0.367i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.929−0.367i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.929−0.367i-0.929 - 0.367i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(393,⋅)\chi_{637} (393, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.929−0.367i)(2,\ 637,\ (\ :1/2),\ -0.929 - 0.367i)

Particular Values

L(1)L(1) ≈\approx 0.396189+2.07871i0.396189 + 2.07871i
L(12)L(\frac12) ≈\approx 0.396189+2.07871i0.396189 + 2.07871i
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+(3.31−1.41i)T 1 + (3.31 - 1.41i)T
good2 1+(−0.651−1.12i)T+(−1+1.73i)T2 1 + (-0.651 - 1.12i)T + (-1 + 1.73i)T^{2}
3 1+(−1.44−2.49i)T+(−1.5+2.59i)T2 1 + (-1.44 - 2.49i)T + (-1.5 + 2.59i)T^{2}
5 1+2.88T+5T2 1 + 2.88T + 5T^{2}
11 1+(−2.95−5.11i)T+(−5.5+9.52i)T2 1 + (-2.95 - 5.11i)T + (-5.5 + 9.52i)T^{2}
17 1+(0.436−0.755i)T+(−8.5−14.7i)T2 1 + (0.436 - 0.755i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.44+2.49i)T+(−9.5−16.4i)T2 1 + (-1.44 + 2.49i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.30+5.72i)T+(−11.5+19.9i)T2 1 + (3.30 + 5.72i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.651+1.12i)T+(−14.5+25.1i)T2 1 + (0.651 + 1.12i)T + (-14.5 + 25.1i)T^{2}
31 1+0.872T+31T2 1 + 0.872T + 31T^{2}
37 1+(0.697+1.20i)T+(−18.5+32.0i)T2 1 + (0.697 + 1.20i)T + (-18.5 + 32.0i)T^{2}
41 1+(−3.75−6.50i)T+(−20.5+35.5i)T2 1 + (-3.75 - 6.50i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.75−4.77i)T+(−21.5−37.2i)T2 1 + (2.75 - 4.77i)T + (-21.5 - 37.2i)T^{2}
47 1−12.3T+47T2 1 - 12.3T + 47T^{2}
53 1−9.60T+53T2 1 - 9.60T + 53T^{2}
59 1+(−3.31+5.74i)T+(−29.5−51.0i)T2 1 + (-3.31 + 5.74i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.88+4.99i)T+(−30.5−52.8i)T2 1 + (-2.88 + 4.99i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.5+0.866i)T+(−33.5+58.0i)T2 1 + (0.5 + 0.866i)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 1−5.76T+73T2 1 - 5.76T + 73T^{2}
79 1−0.605T+79T2 1 - 0.605T + 79T^{2}
83 1−6.63T+83T2 1 - 6.63T + 83T^{2}
89 1+(−4.32−7.48i)T+(−44.5+77.0i)T2 1 + (-4.32 - 7.48i)T + (-44.5 + 77.0i)T^{2}
97 1+(3.88−6.73i)T+(−48.5−84.0i)T2 1 + (3.88 - 6.73i)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

−10.79250531383465555981853067102, −9.929789031761097156703981076889, −9.322566733233334626179026948645, −8.226819865358564604672378990780, −7.43362136098085192960933063189, −6.66788936447869144061971598997, −5.08539115899838524831019224322, −4.31366968624434947174578385894, −4.02865999138294273477824972901, −2.37829516638545847841447354908, 0.950068680988019740981576199563, 2.34319591011078898335992378749, 3.41924764994475015402629261393, 3.89277214277907347779548060582, 5.67644904227778997825071169216, 7.07475470467055880207413879501, 7.55100786847014753191941300454, 8.232448319669320290585132270173, 9.042544795781123294622184580775, 10.52862917633162207505304906989

Graph of the ZZ-function along the critical line