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Properties

Label 2-637-7.2-c1-0-39
Degree 22
Conductor 637637
Sign 0.605−0.795i0.605 - 0.795i
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  + (1.32 − 2.29i)2-s + (−1.19 − 2.07i)3-s + (−2.52 − 4.37i)4-s + (−1.82 + 3.16i)5-s − 6.36·6-s − 8.10·8-s + (−1.37 + 2.37i)9-s + (4.85 + 8.40i)10-s + (−0.327 − 0.567i)11-s + (−6.05 + 10.4i)12-s − 13-s + 8.75·15-s + (−5.70 + 9.88i)16-s + (−1.19 − 2.07i)17-s + (3.63 + 6.30i)18-s + (1.35 − 2.34i)19-s + ⋯
L(s)  = 1  + (0.938 − 1.62i)2-s + (−0.691 − 1.19i)3-s + (−1.26 − 2.18i)4-s + (−0.817 + 1.41i)5-s − 2.59·6-s − 2.86·8-s + (−0.456 + 0.791i)9-s + (1.53 + 2.65i)10-s + (−0.0988 − 0.171i)11-s + (−1.74 + 3.02i)12-s − 0.277·13-s + 2.26·15-s + (−1.42 + 2.47i)16-s + (−0.290 − 0.503i)17-s + (0.857 + 1.48i)18-s + (0.310 − 0.537i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.605−0.795i0.605 - 0.795i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(79,⋅)\chi_{637} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.605−0.795i)(2,\ 637,\ (\ :1/2),\ 0.605 - 0.795i)

Particular Values

L(1)L(1) ≈\approx 0.408789+0.202647i0.408789 + 0.202647i
L(12)L(\frac12) ≈\approx 0.408789+0.202647i0.408789 + 0.202647i
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+T 1 + T
good2 1+(−1.32+2.29i)T+(−1−1.73i)T2 1 + (-1.32 + 2.29i)T + (-1 - 1.73i)T^{2}
3 1+(1.19+2.07i)T+(−1.5+2.59i)T2 1 + (1.19 + 2.07i)T + (-1.5 + 2.59i)T^{2}
5 1+(1.82−3.16i)T+(−2.5−4.33i)T2 1 + (1.82 - 3.16i)T + (-2.5 - 4.33i)T^{2}
11 1+(0.327+0.567i)T+(−5.5+9.52i)T2 1 + (0.327 + 0.567i)T + (-5.5 + 9.52i)T^{2}
17 1+(1.19+2.07i)T+(−8.5+14.7i)T2 1 + (1.19 + 2.07i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.35+2.34i)T+(−9.5−16.4i)T2 1 + (-1.35 + 2.34i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.68−6.37i)T+(−11.5−19.9i)T2 1 + (3.68 - 6.37i)T + (-11.5 - 19.9i)T^{2}
29 1+0.208T+29T2 1 + 0.208T + 29T^{2}
31 1+(−0.568−0.984i)T+(−15.5+26.8i)T2 1 + (-0.568 - 0.984i)T + (-15.5 + 26.8i)T^{2}
37 1+(−3.72+6.44i)T+(−18.5−32.0i)T2 1 + (-3.72 + 6.44i)T + (-18.5 - 32.0i)T^{2}
41 1+10.2T+41T2 1 + 10.2T + 41T^{2}
43 1+3.10T+43T2 1 + 3.10T + 43T^{2}
47 1+(2.30−3.98i)T+(−23.5−40.7i)T2 1 + (2.30 - 3.98i)T + (-23.5 - 40.7i)T^{2}
53 1+(2.62+4.55i)T+(−26.5+45.8i)T2 1 + (2.62 + 4.55i)T + (-26.5 + 45.8i)T^{2}
59 1+(4.12+7.15i)T+(−29.5+51.0i)T2 1 + (4.12 + 7.15i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.948+1.64i)T+(−30.5−52.8i)T2 1 + (-0.948 + 1.64i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.44−11.1i)T+(−33.5+58.0i)T2 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2}
71 1+6.75T+71T2 1 + 6.75T + 71T^{2}
73 1+(6.26+10.8i)T+(−36.5+63.2i)T2 1 + (6.26 + 10.8i)T + (-36.5 + 63.2i)T^{2}
79 1+(−0.759+1.31i)T+(−39.5−68.4i)T2 1 + (-0.759 + 1.31i)T + (-39.5 - 68.4i)T^{2}
83 1−15.7T+83T2 1 - 15.7T + 83T^{2}
89 1+(−7.40+12.8i)T+(−44.5−77.0i)T2 1 + (-7.40 + 12.8i)T + (-44.5 - 77.0i)T^{2}
97 1+10.0T+97T2 1 + 10.0T + 97T^{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

−10.37205698153630817217161212774, −9.440285385431193561552392757425, −7.81455459710515191297329728054, −6.94997380370416874544885279001, −6.10945373788534304965244059756, −5.07571938015031445044820164257, −3.76513866867843848473978182481, −2.92005616738070379685673315941, −1.82461341601746379129452685230, −0.19524746124879706762616610652, 3.64314792404633435687460281059, 4.43055169186965200662577830710, 4.84600175098474077328786385672, 5.63950365973091143030525576815, 6.59877199414082053929289835024, 7.902172242616274661162761795685, 8.378965389340972895748740422468, 9.298603896222744139248339022057, 10.32793182589722236471608852695, 11.72892229826304670095027978906

Graph of the ZZ-function along the critical line