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Properties

Label 2-637-13.10-c1-0-6
Degree 22
Conductor 637637
Sign −0.964+0.265i-0.964 + 0.265i
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  + (−0.395 + 0.228i)2-s + (1.39 + 2.41i)3-s + (−0.895 + 1.55i)4-s + 0.456i·5-s + (−1.10 − 0.637i)6-s − 1.73i·8-s + (−2.39 + 4.14i)9-s + (−0.104 − 0.180i)10-s + (−3.39 + 1.96i)11-s − 5·12-s + (−3.5 − 0.866i)13-s + (−1.10 + 0.637i)15-s + (−1.39 − 2.41i)16-s + (1.5 − 2.59i)17-s − 2.18i·18-s + (−1.18 − 0.685i)19-s + ⋯
L(s)  = 1  + (−0.279 + 0.161i)2-s + (0.805 + 1.39i)3-s + (−0.447 + 0.775i)4-s + 0.204i·5-s + (−0.450 − 0.260i)6-s − 0.612i·8-s + (−0.798 + 1.38i)9-s + (−0.0330 − 0.0571i)10-s + (−1.02 + 0.591i)11-s − 1.44·12-s + (−0.970 − 0.240i)13-s + (−0.285 + 0.164i)15-s + (−0.348 − 0.604i)16-s + (0.363 − 0.630i)17-s − 0.515i·18-s + (−0.272 − 0.157i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.964+0.265i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.964+0.265i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.964+0.265i-0.964 + 0.265i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(491,⋅)\chi_{637} (491, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.964+0.265i)(2,\ 637,\ (\ :1/2),\ -0.964 + 0.265i)

Particular Values

L(1)L(1) ≈\approx 0.139667−1.03522i0.139667 - 1.03522i
L(12)L(\frac12) ≈\approx 0.139667−1.03522i0.139667 - 1.03522i
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+(3.5+0.866i)T 1 + (3.5 + 0.866i)T
good2 1+(0.395−0.228i)T+(1−1.73i)T2 1 + (0.395 - 0.228i)T + (1 - 1.73i)T^{2}
3 1+(−1.39−2.41i)T+(−1.5+2.59i)T2 1 + (-1.39 - 2.41i)T + (-1.5 + 2.59i)T^{2}
5 1−0.456iT−5T2 1 - 0.456iT - 5T^{2}
11 1+(3.39−1.96i)T+(5.5−9.52i)T2 1 + (3.39 - 1.96i)T + (5.5 - 9.52i)T^{2}
17 1+(−1.5+2.59i)T+(−8.5−14.7i)T2 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.18+0.685i)T+(9.5+16.4i)T2 1 + (1.18 + 0.685i)T + (9.5 + 16.4i)T^{2}
23 1+(−0.791−1.37i)T+(−11.5+19.9i)T2 1 + (-0.791 - 1.37i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.39−5.88i)T+(−14.5+25.1i)T2 1 + (-3.39 - 5.88i)T + (-14.5 + 25.1i)T^{2}
31 1−8.66iT−31T2 1 - 8.66iT - 31T^{2}
37 1+(−6+3.46i)T+(18.5−32.0i)T2 1 + (-6 + 3.46i)T + (18.5 - 32.0i)T^{2}
41 1+(6.79−3.92i)T+(20.5−35.5i)T2 1 + (6.79 - 3.92i)T + (20.5 - 35.5i)T^{2}
43 1+(4.68−8.11i)T+(−21.5−37.2i)T2 1 + (4.68 - 8.11i)T + (-21.5 - 37.2i)T^{2}
47 1+9.57iT−47T2 1 + 9.57iT - 47T^{2}
53 1−6.16T+53T2 1 - 6.16T + 53T^{2}
59 1+(10.6+6.15i)T+(29.5+51.0i)T2 1 + (10.6 + 6.15i)T + (29.5 + 51.0i)T^{2}
61 1+(7.37−12.7i)T+(−30.5−52.8i)T2 1 + (7.37 - 12.7i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.87−2.23i)T+(33.5−58.0i)T2 1 + (3.87 - 2.23i)T + (33.5 - 58.0i)T^{2}
71 1+(−3.79−2.18i)T+(35.5+61.4i)T2 1 + (-3.79 - 2.18i)T + (35.5 + 61.4i)T^{2}
73 1+3.46iT−73T2 1 + 3.46iT - 73T^{2}
79 1+6T+79T2 1 + 6T + 79T^{2}
83 1+7.02iT−83T2 1 + 7.02iT - 83T^{2}
89 1+(−13.9+8.07i)T+(44.5−77.0i)T2 1 + (-13.9 + 8.07i)T + (44.5 - 77.0i)T^{2}
97 1+(−6.31−3.64i)T+(48.5+84.0i)T2 1 + (-6.31 - 3.64i)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

−10.51282413827887361165080505597, −10.13464250534185857531211473455, −9.258714560175922309285970252515, −8.627512684767130511465316451076, −7.74877425168365594260237677889, −6.98189212705122811679862003759, −4.96206817363485237646196803907, −4.74073964159259720591452148693, −3.29498762492748357425125107314, −2.77811873909924684767672166968, 0.55432840540967627536145525108, 1.92137794749078930091982327511, 2.84146579155933846267424105971, 4.56667528476319610584813868021, 5.72595265434186874292787510418, 6.58528229148462483192431559472, 7.78203428673521880109879486901, 8.233523347732689336078091939567, 9.098994891542534246660322649804, 10.00621174066112442862171343157

Graph of the ZZ-function along the critical line