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Properties

Label 2-637-91.9-c1-0-22
Degree 22
Conductor 637637
Sign 0.471+0.882i0.471 + 0.882i
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  + (0.289 + 0.502i)2-s − 1.89·3-s + (0.831 − 1.44i)4-s + (−0.736 + 1.27i)5-s + (−0.548 − 0.950i)6-s + 2.12·8-s + 0.579·9-s − 0.854·10-s + 0.579·11-s + (−1.57 + 2.72i)12-s + (−0.128 − 3.60i)13-s + (1.39 − 2.41i)15-s + (−1.04 − 1.81i)16-s + (0.598 − 1.03i)17-s + (0.168 + 0.291i)18-s − 0.460·19-s + ⋯
L(s)  = 1  + (0.204 + 0.355i)2-s − 1.09·3-s + (0.415 − 0.720i)4-s + (−0.329 + 0.570i)5-s + (−0.223 − 0.387i)6-s + 0.751·8-s + 0.193·9-s − 0.270·10-s + 0.174·11-s + (−0.454 + 0.787i)12-s + (−0.0357 − 0.999i)13-s + (0.359 − 0.623i)15-s + (−0.261 − 0.453i)16-s + (0.145 − 0.251i)17-s + (0.0396 + 0.0686i)18-s − 0.105·19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.471+0.882i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.471+0.882i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.471+0.882i0.471 + 0.882i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(373,⋅)\chi_{637} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.471+0.882i)(2,\ 637,\ (\ :1/2),\ 0.471 + 0.882i)

Particular Values

L(1)L(1) ≈\approx 0.861933−0.516829i0.861933 - 0.516829i
L(12)L(\frac12) ≈\approx 0.861933−0.516829i0.861933 - 0.516829i
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+(0.128+3.60i)T 1 + (0.128 + 3.60i)T
good2 1+(−0.289−0.502i)T+(−1+1.73i)T2 1 + (-0.289 - 0.502i)T + (-1 + 1.73i)T^{2}
3 1+1.89T+3T2 1 + 1.89T + 3T^{2}
5 1+(0.736−1.27i)T+(−2.5−4.33i)T2 1 + (0.736 - 1.27i)T + (-2.5 - 4.33i)T^{2}
11 1−0.579T+11T2 1 - 0.579T + 11T^{2}
17 1+(−0.598+1.03i)T+(−8.5−14.7i)T2 1 + (-0.598 + 1.03i)T + (-8.5 - 14.7i)T^{2}
19 1+0.460T+19T2 1 + 0.460T + 19T^{2}
23 1+(1.18+2.05i)T+(−11.5+19.9i)T2 1 + (1.18 + 2.05i)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+(2.22+3.84i)T+(−15.5+26.8i)T2 1 + (2.22 + 3.84i)T + (-15.5 + 26.8i)T^{2}
37 1+(4.58+7.93i)T+(−18.5+32.0i)T2 1 + (4.58 + 7.93i)T + (-18.5 + 32.0i)T^{2}
41 1+(−2.00+3.47i)T+(−20.5−35.5i)T2 1 + (-2.00 + 3.47i)T + (-20.5 - 35.5i)T^{2}
43 1+(4.02+6.97i)T+(−21.5+37.2i)T2 1 + (4.02 + 6.97i)T + (-21.5 + 37.2i)T^{2}
47 1+(−5.75+9.97i)T+(−23.5−40.7i)T2 1 + (-5.75 + 9.97i)T + (-23.5 - 40.7i)T^{2}
53 1+(−4.69−8.13i)T+(−26.5+45.8i)T2 1 + (-4.69 - 8.13i)T + (-26.5 + 45.8i)T^{2}
59 1+(0.120−0.208i)T+(−29.5−51.0i)T2 1 + (0.120 - 0.208i)T + (-29.5 - 51.0i)T^{2}
61 1−7.72T+61T2 1 - 7.72T + 61T^{2}
67 1+1.44T+67T2 1 + 1.44T + 67T^{2}
71 1+(−6.25−10.8i)T+(−35.5+61.4i)T2 1 + (-6.25 - 10.8i)T + (-35.5 + 61.4i)T^{2}
73 1+(−1.84−3.19i)T+(−36.5+63.2i)T2 1 + (-1.84 - 3.19i)T + (-36.5 + 63.2i)T^{2}
79 1+(8.03−13.9i)T+(−39.5−68.4i)T2 1 + (8.03 - 13.9i)T + (-39.5 - 68.4i)T^{2}
83 1+15.4T+83T2 1 + 15.4T + 83T^{2}
89 1+(1.24+2.15i)T+(−44.5+77.0i)T2 1 + (1.24 + 2.15i)T + (-44.5 + 77.0i)T^{2}
97 1+(7.82+13.5i)T+(−48.5+84.0i)T2 1 + (7.82 + 13.5i)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

−10.54087804395691024178131794045, −9.998229171609025365026002359339, −8.593327061210134546466172361920, −7.36615536854117370901645667668, −6.78864333083529584085502565648, −5.71893162263153022911162748028, −5.39063775777190702686948514594, −4.03582847618678352470385246225, −2.48272078879581929991038210113, −0.61498371685413355659004989304, 1.47003209809559148518823047527, 3.09820664560941021363336372774, 4.32324228638797322867602781170, 5.03485345398143778064823912804, 6.33010015300861960803316643531, 6.98246751193299329179854128527, 8.153358642086044038870938033041, 8.880668617204004762065250086422, 10.17147538091222620459639221431, 11.03984552273993751628501485457

Graph of the ZZ-function along the critical line