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Properties

Label 2-637-13.9-c1-0-7
Degree 22
Conductor 637637
Sign −0.727−0.686i-0.727 - 0.686i
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 + (−0.946 + 1.63i)3-s + (0.831 + 1.44i)4-s − 1.47·5-s + (0.548 + 0.950i)6-s + 2.12·8-s + (−0.289 − 0.502i)9-s + (−0.427 + 0.739i)10-s + (−0.289 + 0.502i)11-s − 3.14·12-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 − 0.336·18-s + (−0.230 − 0.399i)19-s + ⋯
L(s)  = 1  + (0.204 − 0.355i)2-s + (−0.546 + 0.946i)3-s + (0.415 + 0.720i)4-s − 0.659·5-s + (0.223 + 0.387i)6-s + 0.751·8-s + (−0.0966 − 0.167i)9-s + (−0.135 + 0.233i)10-s + (−0.0874 + 0.151i)11-s − 0.908·12-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 − 0.0792·18-s + (−0.0528 − 0.0915i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.727−0.686i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.727−0.686i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.727−0.686i-0.727 - 0.686i
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), −0.727−0.686i)(2,\ 637,\ (\ :1/2),\ -0.727 - 0.686i)

Particular Values

L(1)L(1) ≈\approx 0.399085+1.00493i0.399085 + 1.00493i
L(12)L(\frac12) ≈\approx 0.399085+1.00493i0.399085 + 1.00493i
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+(0.946−1.63i)T+(−1.5−2.59i)T2 1 + (0.946 - 1.63i)T + (-1.5 - 2.59i)T^{2}
5 1+1.47T+5T2 1 + 1.47T + 5T^{2}
11 1+(0.289−0.502i)T+(−5.5−9.52i)T2 1 + (0.289 - 0.502i)T + (-5.5 - 9.52i)T^{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+(0.230+0.399i)T+(−9.5+16.4i)T2 1 + (0.230 + 0.399i)T + (-9.5 + 16.4i)T^{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+4.44T+31T2 1 + 4.44T + 31T^{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 1−11.5T+47T2 1 - 11.5T + 47T^{2}
53 1+9.39T+53T2 1 + 9.39T + 53T^{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+(−3.86−6.69i)T+(−30.5+52.8i)T2 1 + (-3.86 - 6.69i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.724+1.25i)T+(−33.5−58.0i)T2 1 + (-0.724 + 1.25i)T + (-33.5 - 58.0i)T^{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 1−3.69T+73T2 1 - 3.69T + 73T^{2}
79 1−16.0T+79T2 1 - 16.0T + 79T^{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

−11.03837529637756488341663371640, −10.26441613989497081990904327000, −9.357422373832373561288348639024, −8.239559601105555414767950662217, −7.42354659572144616639578719615, −6.47460325099959838699669638347, −5.09630285376624071631168074955, −4.23106745418238047705361165036, −3.60101682255721948678521297275, −2.11912395278826651169602815346, 0.57451372504531519107677728228, 1.94674195769371218729679271588, 3.61092968301942141433974748126, 5.02601188429501087547840907485, 5.87049528447841512235156233183, 6.63431067836365669826930119731, 7.43126691754633802862319688094, 8.111024057189082023278897882429, 9.412193741088264566844083531002, 10.62569210530784445695300894992

Graph of the ZZ-function along the critical line