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Properties

Label 2-637-91.81-c1-0-22
Degree 22
Conductor 637637
Sign 0.993+0.110i0.993 + 0.110i
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  + (0.134 − 0.232i)2-s + 1.14·3-s + (0.964 + 1.66i)4-s + (−1.28 − 2.21i)5-s + (0.153 − 0.265i)6-s + 1.05·8-s − 1.69·9-s − 0.686·10-s + 3.94·11-s + (1.10 + 1.90i)12-s + (3.15 + 1.74i)13-s + (−1.46 − 2.53i)15-s + (−1.78 + 3.09i)16-s + (0.392 + 0.679i)17-s + (−0.227 + 0.393i)18-s + 7.49·19-s + ⋯
L(s)  = 1  + (0.0947 − 0.164i)2-s + 0.659·3-s + (0.482 + 0.834i)4-s + (−0.572 − 0.992i)5-s + (0.0625 − 0.108i)6-s + 0.372·8-s − 0.564·9-s − 0.217·10-s + 1.18·11-s + (0.318 + 0.550i)12-s + (0.874 + 0.484i)13-s + (−0.378 − 0.654i)15-s + (−0.446 + 0.773i)16-s + (0.0952 + 0.164i)17-s + (−0.0535 + 0.0926i)18-s + 1.71·19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.993+0.110i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.993+0.110i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.993+0.110i0.993 + 0.110i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(263,⋅)\chi_{637} (263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.993+0.110i)(2,\ 637,\ (\ :1/2),\ 0.993 + 0.110i)

Particular Values

L(1)L(1) ≈\approx 2.10256−0.116852i2.10256 - 0.116852i
L(12)L(\frac12) ≈\approx 2.10256−0.116852i2.10256 - 0.116852i
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+(−3.15−1.74i)T 1 + (-3.15 - 1.74i)T
good2 1+(−0.134+0.232i)T+(−1−1.73i)T2 1 + (-0.134 + 0.232i)T + (-1 - 1.73i)T^{2}
3 1−1.14T+3T2 1 - 1.14T + 3T^{2}
5 1+(1.28+2.21i)T+(−2.5+4.33i)T2 1 + (1.28 + 2.21i)T + (-2.5 + 4.33i)T^{2}
11 1−3.94T+11T2 1 - 3.94T + 11T^{2}
17 1+(−0.392−0.679i)T+(−8.5+14.7i)T2 1 + (-0.392 - 0.679i)T + (-8.5 + 14.7i)T^{2}
19 1−7.49T+19T2 1 - 7.49T + 19T^{2}
23 1+(−3.97+6.88i)T+(−11.5−19.9i)T2 1 + (-3.97 + 6.88i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.17+2.03i)T+(−14.5+25.1i)T2 1 + (1.17 + 2.03i)T + (-14.5 + 25.1i)T^{2}
31 1+(1.27−2.21i)T+(−15.5−26.8i)T2 1 + (1.27 - 2.21i)T + (-15.5 - 26.8i)T^{2}
37 1+(3.37−5.85i)T+(−18.5−32.0i)T2 1 + (3.37 - 5.85i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.21+2.11i)T+(−20.5+35.5i)T2 1 + (1.21 + 2.11i)T + (-20.5 + 35.5i)T^{2}
43 1+(−1.12+1.94i)T+(−21.5−37.2i)T2 1 + (-1.12 + 1.94i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.658−1.14i)T+(−23.5+40.7i)T2 1 + (-0.658 - 1.14i)T + (-23.5 + 40.7i)T^{2}
53 1+(4.63−8.03i)T+(−26.5−45.8i)T2 1 + (4.63 - 8.03i)T + (-26.5 - 45.8i)T^{2}
59 1+(4.48+7.76i)T+(−29.5+51.0i)T2 1 + (4.48 + 7.76i)T + (-29.5 + 51.0i)T^{2}
61 1+9.44T+61T2 1 + 9.44T + 61T^{2}
67 1+1.35T+67T2 1 + 1.35T + 67T^{2}
71 1+(6.15−10.6i)T+(−35.5−61.4i)T2 1 + (6.15 - 10.6i)T + (-35.5 - 61.4i)T^{2}
73 1+(−0.384+0.665i)T+(−36.5−63.2i)T2 1 + (-0.384 + 0.665i)T + (-36.5 - 63.2i)T^{2}
79 1+(3.09+5.36i)T+(−39.5+68.4i)T2 1 + (3.09 + 5.36i)T + (-39.5 + 68.4i)T^{2}
83 1−1.07T+83T2 1 - 1.07T + 83T^{2}
89 1+(−3.83+6.63i)T+(−44.5−77.0i)T2 1 + (-3.83 + 6.63i)T + (-44.5 - 77.0i)T^{2}
97 1+(1.18−2.05i)T+(−48.5−84.0i)T2 1 + (1.18 - 2.05i)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

−10.79705862303205479255916628044, −9.267991484368935091323252530342, −8.768593540217079824458690079728, −8.146541540915947626765274099141, −7.20778509067091064825312967349, −6.18031554966600123735391862101, −4.72354286717446626936206168784, −3.77938695102700815279686275971, −2.99022730877226756757896403161, −1.39073571632842701878859956515, 1.41306680102806652831764531541, 3.04274769886717482463794801712, 3.60753819574254890310851901149, 5.31341062796277158378853398467, 6.13791271734633170549377549826, 7.17342794707318778613847795629, 7.68479973598498888600173975747, 9.033875640800908777216968937787, 9.589601312280158847178103287551, 10.79425901372035853135569831530

Graph of the ZZ-function along the critical line