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Properties

Label 2-637-13.3-c1-0-14
Degree 22
Conductor 637637
Sign −0.384−0.922i-0.384 - 0.922i
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  + (0.760 + 1.31i)2-s + (1.06 + 1.84i)3-s + (−0.156 + 0.270i)4-s + 0.589·5-s + (−1.61 + 2.80i)6-s + 2.56·8-s + (−0.760 + 1.31i)9-s + (0.448 + 0.776i)10-s + (−0.760 − 1.31i)11-s − 0.664·12-s + (3.32 + 1.39i)13-s + (0.626 + 1.08i)15-s + (2.26 + 3.92i)16-s + (−2.39 + 4.15i)17-s − 2.31·18-s + (−0.841 + 1.45i)19-s + ⋯
L(s)  = 1  + (0.537 + 0.931i)2-s + (0.613 + 1.06i)3-s + (−0.0781 + 0.135i)4-s + 0.263·5-s + (−0.660 + 1.14i)6-s + 0.907·8-s + (−0.253 + 0.439i)9-s + (0.141 + 0.245i)10-s + (−0.229 − 0.397i)11-s − 0.191·12-s + (0.922 + 0.386i)13-s + (0.161 + 0.280i)15-s + (0.565 + 0.980i)16-s + (−0.581 + 1.00i)17-s − 0.545·18-s + (−0.193 + 0.334i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.384−0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.384−0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.384 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.384−0.922i-0.384 - 0.922i
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), −0.384−0.922i)(2,\ 637,\ (\ :1/2),\ -0.384 - 0.922i)

Particular Values

L(1)L(1) ≈\approx 1.50939+2.26506i1.50939 + 2.26506i
L(12)L(\frac12) ≈\approx 1.50939+2.26506i1.50939 + 2.26506i
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+(−3.32−1.39i)T 1 + (-3.32 - 1.39i)T
good2 1+(−0.760−1.31i)T+(−1+1.73i)T2 1 + (-0.760 - 1.31i)T + (-1 + 1.73i)T^{2}
3 1+(−1.06−1.84i)T+(−1.5+2.59i)T2 1 + (-1.06 - 1.84i)T + (-1.5 + 2.59i)T^{2}
5 1−0.589T+5T2 1 - 0.589T + 5T^{2}
11 1+(0.760+1.31i)T+(−5.5+9.52i)T2 1 + (0.760 + 1.31i)T + (-5.5 + 9.52i)T^{2}
17 1+(2.39−4.15i)T+(−8.5−14.7i)T2 1 + (2.39 - 4.15i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.841−1.45i)T+(−9.5−16.4i)T2 1 + (0.841 - 1.45i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.886+1.53i)T+(−11.5+19.9i)T2 1 + (0.886 + 1.53i)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+6.08T+31T2 1 + 6.08T + 31T^{2}
37 1+(0.704+1.22i)T+(−18.5+32.0i)T2 1 + (0.704 + 1.22i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.677+1.17i)T+(−20.5+35.5i)T2 1 + (0.677 + 1.17i)T + (-20.5 + 35.5i)T^{2}
43 1+(−5.77+10.0i)T+(−21.5−37.2i)T2 1 + (-5.77 + 10.0i)T + (-21.5 - 37.2i)T^{2}
47 1+0.464T+47T2 1 + 0.464T + 47T^{2}
53 1−8.24T+53T2 1 - 8.24T + 53T^{2}
59 1+(−5.93+10.2i)T+(−29.5−51.0i)T2 1 + (-5.93 + 10.2i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.24−2.14i)T+(−30.5−52.8i)T2 1 + (1.24 - 2.14i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.78−6.55i)T+(−33.5+58.0i)T2 1 + (-3.78 - 6.55i)T + (-33.5 + 58.0i)T^{2}
71 1+(3.30−5.71i)T+(−35.5−61.4i)T2 1 + (3.30 - 5.71i)T + (-35.5 - 61.4i)T^{2}
73 1+16.3T+73T2 1 + 16.3T + 73T^{2}
79 1+14.9T+79T2 1 + 14.9T + 79T^{2}
83 1−10.1T+83T2 1 - 10.1T + 83T^{2}
89 1+(8.24+14.2i)T+(−44.5+77.0i)T2 1 + (8.24 + 14.2i)T + (-44.5 + 77.0i)T^{2}
97 1+(−0.486+0.843i)T+(−48.5−84.0i)T2 1 + (-0.486 + 0.843i)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

−10.59642131982415613345034539959, −10.05005414095369143970241146965, −8.936365249817399328127448105663, −8.336257165473213422778812221928, −7.21554929619358064909680851743, −6.10832746671092656846130164387, −5.55766558227228067816851031032, −4.18162897725555001876742784698, −3.80972534675549788431734296624, −2.01061848449569760489930310482, 1.42568431320038204824636494391, 2.35079628947152365897669539918, 3.26574645031640555276931114328, 4.46254209251003209617155633425, 5.67747042755073640721513107154, 7.01551029957082355296052949411, 7.54022606915036752716598066733, 8.511427369344485678857459171185, 9.492894849439807616682533248596, 10.60032147713254548156925799165

Graph of the ZZ-function along the critical line