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Properties

Label 2-637-91.54-c1-0-27
Degree 22
Conductor 637637
Sign 0.777+0.628i0.777 + 0.628i
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  + (0.653 − 0.653i)2-s + (−0.988 − 0.570i)3-s + 1.14i·4-s + (3.82 − 1.02i)5-s + (−1.01 + 0.272i)6-s + (2.05 + 2.05i)8-s + (−0.848 − 1.47i)9-s + (1.82 − 3.16i)10-s + (−2.03 + 0.544i)11-s + (0.654 − 1.13i)12-s + (3.41 − 1.16i)13-s + (−4.36 − 1.16i)15-s + 0.391·16-s + 3.17·17-s + (−1.51 − 0.405i)18-s + (−0.302 + 1.12i)19-s + ⋯
L(s)  = 1  + (0.461 − 0.461i)2-s + (−0.570 − 0.329i)3-s + 0.573i·4-s + (1.71 − 0.458i)5-s + (−0.415 + 0.111i)6-s + (0.726 + 0.726i)8-s + (−0.282 − 0.490i)9-s + (0.578 − 1.00i)10-s + (−0.613 + 0.164i)11-s + (0.188 − 0.327i)12-s + (0.946 − 0.324i)13-s + (−1.12 − 0.301i)15-s + 0.0979·16-s + 0.769·17-s + (−0.357 − 0.0956i)18-s + (−0.0693 + 0.258i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.777+0.628i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.777+0.628i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.777+0.628i0.777 + 0.628i
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), 0.777+0.628i)(2,\ 637,\ (\ :1/2),\ 0.777 + 0.628i)

Particular Values

L(1)L(1) ≈\approx 2.00589−0.709057i2.00589 - 0.709057i
L(12)L(\frac12) ≈\approx 2.00589−0.709057i2.00589 - 0.709057i
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+(−3.41+1.16i)T 1 + (-3.41 + 1.16i)T
good2 1+(−0.653+0.653i)T−2iT2 1 + (-0.653 + 0.653i)T - 2iT^{2}
3 1+(0.988+0.570i)T+(1.5+2.59i)T2 1 + (0.988 + 0.570i)T + (1.5 + 2.59i)T^{2}
5 1+(−3.82+1.02i)T+(4.33−2.5i)T2 1 + (-3.82 + 1.02i)T + (4.33 - 2.5i)T^{2}
11 1+(2.03−0.544i)T+(9.52−5.5i)T2 1 + (2.03 - 0.544i)T + (9.52 - 5.5i)T^{2}
17 1−3.17T+17T2 1 - 3.17T + 17T^{2}
19 1+(0.302−1.12i)T+(−16.4−9.5i)T2 1 + (0.302 - 1.12i)T + (-16.4 - 9.5i)T^{2}
23 1−3.76iT−23T2 1 - 3.76iT - 23T^{2}
29 1+(0.584+1.01i)T+(−14.5+25.1i)T2 1 + (0.584 + 1.01i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.30+4.88i)T+(−26.8−15.5i)T2 1 + (-1.30 + 4.88i)T + (-26.8 - 15.5i)T^{2}
37 1+(−3.12−3.12i)T+37iT2 1 + (-3.12 - 3.12i)T + 37iT^{2}
41 1+(−1.85+6.93i)T+(−35.5−20.5i)T2 1 + (-1.85 + 6.93i)T + (-35.5 - 20.5i)T^{2}
43 1+(1.91+1.10i)T+(21.5+37.2i)T2 1 + (1.91 + 1.10i)T + (21.5 + 37.2i)T^{2}
47 1+(3.00+11.2i)T+(−40.7+23.5i)T2 1 + (3.00 + 11.2i)T + (-40.7 + 23.5i)T^{2}
53 1+(2.44+4.23i)T+(−26.5+45.8i)T2 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2}
59 1+(−0.173+0.173i)T−59iT2 1 + (-0.173 + 0.173i)T - 59iT^{2}
61 1+(10.7−6.18i)T+(30.5−52.8i)T2 1 + (10.7 - 6.18i)T + (30.5 - 52.8i)T^{2}
67 1+(−2.66−9.95i)T+(−58.0+33.5i)T2 1 + (-2.66 - 9.95i)T + (-58.0 + 33.5i)T^{2}
71 1+(−1.60−5.98i)T+(−61.4+35.5i)T2 1 + (-1.60 - 5.98i)T + (-61.4 + 35.5i)T^{2}
73 1+(−4.75−1.27i)T+(63.2+36.5i)T2 1 + (-4.75 - 1.27i)T + (63.2 + 36.5i)T^{2}
79 1+(−1.34+2.33i)T+(−39.5−68.4i)T2 1 + (-1.34 + 2.33i)T + (-39.5 - 68.4i)T^{2}
83 1+(−3.31−3.31i)T+83iT2 1 + (-3.31 - 3.31i)T + 83iT^{2}
89 1+(4.91−4.91i)T−89iT2 1 + (4.91 - 4.91i)T - 89iT^{2}
97 1+(15.8−4.24i)T+(84.0−48.5i)T2 1 + (15.8 - 4.24i)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

−10.56875041127182467826077768617, −9.762086639668070617193954590921, −8.819992684449529719318146175482, −7.910546400501643727024582304116, −6.66236302783008192290593447297, −5.67813074171398156082239737789, −5.27115354419690166868316883103, −3.75854837299399375234557323570, −2.56447866009545757515211163742, −1.36727586594978507851610199146, 1.50220814380227645265497368635, 2.85111529516788332149216230814, 4.64457806164946105350975597140, 5.37223415759108494350198852610, 6.11044567155878617839509292876, 6.52277094605627721228788689580, 7.917144403414976944796898401518, 9.244996401146003339383368883312, 9.938222689934786718185303524377, 10.82945780129264237114006894637

Graph of the ZZ-function along the critical line