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Properties

Label 2-637-91.54-c1-0-15
Degree 22
Conductor 637637
Sign 0.995−0.0982i0.995 - 0.0982i
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  + (−1.18 + 1.18i)2-s + (0.552 + 0.318i)3-s − 0.809i·4-s + (−1.94 + 0.520i)5-s + (−1.03 + 0.276i)6-s + (−1.41 − 1.41i)8-s + (−1.29 − 2.24i)9-s + (1.68 − 2.91i)10-s + (−0.948 + 0.254i)11-s + (0.258 − 0.446i)12-s + (1.60 − 3.22i)13-s + (−1.23 − 0.331i)15-s + 4.96·16-s + 5.98·17-s + (4.19 + 1.12i)18-s + (0.726 − 2.71i)19-s + ⋯
L(s)  = 1  + (−0.838 + 0.838i)2-s + (0.318 + 0.184i)3-s − 0.404i·4-s + (−0.868 + 0.232i)5-s + (−0.421 + 0.112i)6-s + (−0.498 − 0.498i)8-s + (−0.432 − 0.748i)9-s + (0.532 − 0.922i)10-s + (−0.285 + 0.0766i)11-s + (0.0744 − 0.129i)12-s + (0.446 − 0.894i)13-s + (−0.319 − 0.0856i)15-s + 1.24·16-s + 1.45·17-s + (0.989 + 0.265i)18-s + (0.166 − 0.622i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.995−0.0982i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0982i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.995−0.0982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.995−0.0982i0.995 - 0.0982i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(509,⋅)\chi_{637} (509, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.995−0.0982i)(2,\ 637,\ (\ :1/2),\ 0.995 - 0.0982i)

Particular Values

L(1)L(1) ≈\approx 0.713763+0.0351334i0.713763 + 0.0351334i
L(12)L(\frac12) ≈\approx 0.713763+0.0351334i0.713763 + 0.0351334i
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+(−1.60+3.22i)T 1 + (-1.60 + 3.22i)T
good2 1+(1.18−1.18i)T−2iT2 1 + (1.18 - 1.18i)T - 2iT^{2}
3 1+(−0.552−0.318i)T+(1.5+2.59i)T2 1 + (-0.552 - 0.318i)T + (1.5 + 2.59i)T^{2}
5 1+(1.94−0.520i)T+(4.33−2.5i)T2 1 + (1.94 - 0.520i)T + (4.33 - 2.5i)T^{2}
11 1+(0.948−0.254i)T+(9.52−5.5i)T2 1 + (0.948 - 0.254i)T + (9.52 - 5.5i)T^{2}
17 1−5.98T+17T2 1 - 5.98T + 17T^{2}
19 1+(−0.726+2.71i)T+(−16.4−9.5i)T2 1 + (-0.726 + 2.71i)T + (-16.4 - 9.5i)T^{2}
23 1−2.98iT−23T2 1 - 2.98iT - 23T^{2}
29 1+(−3.65−6.33i)T+(−14.5+25.1i)T2 1 + (-3.65 - 6.33i)T + (-14.5 + 25.1i)T^{2}
31 1+(−2.23+8.34i)T+(−26.8−15.5i)T2 1 + (-2.23 + 8.34i)T + (-26.8 - 15.5i)T^{2}
37 1+(3.39+3.39i)T+37iT2 1 + (3.39 + 3.39i)T + 37iT^{2}
41 1+(−0.886+3.30i)T+(−35.5−20.5i)T2 1 + (-0.886 + 3.30i)T + (-35.5 - 20.5i)T^{2}
43 1+(0.748+0.432i)T+(21.5+37.2i)T2 1 + (0.748 + 0.432i)T + (21.5 + 37.2i)T^{2}
47 1+(−0.794−2.96i)T+(−40.7+23.5i)T2 1 + (-0.794 - 2.96i)T + (-40.7 + 23.5i)T^{2}
53 1+(3.16+5.47i)T+(−26.5+45.8i)T2 1 + (3.16 + 5.47i)T + (-26.5 + 45.8i)T^{2}
59 1+(−0.359+0.359i)T−59iT2 1 + (-0.359 + 0.359i)T - 59iT^{2}
61 1+(−11.2+6.51i)T+(30.5−52.8i)T2 1 + (-11.2 + 6.51i)T + (30.5 - 52.8i)T^{2}
67 1+(−0.221−0.827i)T+(−58.0+33.5i)T2 1 + (-0.221 - 0.827i)T + (-58.0 + 33.5i)T^{2}
71 1+(3.01+11.2i)T+(−61.4+35.5i)T2 1 + (3.01 + 11.2i)T + (-61.4 + 35.5i)T^{2}
73 1+(1.40+0.377i)T+(63.2+36.5i)T2 1 + (1.40 + 0.377i)T + (63.2 + 36.5i)T^{2}
79 1+(−5.80+10.0i)T+(−39.5−68.4i)T2 1 + (-5.80 + 10.0i)T + (-39.5 - 68.4i)T^{2}
83 1+(−1.23−1.23i)T+83iT2 1 + (-1.23 - 1.23i)T + 83iT^{2}
89 1+(5.67−5.67i)T−89iT2 1 + (5.67 - 5.67i)T - 89iT^{2}
97 1+(−12.0+3.23i)T+(84.0−48.5i)T2 1 + (-12.0 + 3.23i)T + (84.0 - 48.5i)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

−10.29268674226072336150361126355, −9.505604128015299108113014046261, −8.677541696544061436437275668664, −7.898625464652720683541711098712, −7.42900572198446193350530195912, −6.32382185273409387981107166052, −5.38616893266321006500408656100, −3.68642725644974691687326264841, −3.14533246522692643076968967908, −0.58069891695821916258969007881, 1.20706068642651251545832153647, 2.52773460287174272471135043245, 3.60340444658368558162374376072, 4.94173385755474189528677323201, 6.06680484371880923756750602545, 7.47700421931175036470366442013, 8.330041101946419065681402596102, 8.616413151492541140182267881647, 9.908445410546681798341935893750, 10.42020971235078067320656638644

Graph of the ZZ-function along the critical line