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Properties

Label 2-637-91.74-c1-0-34
Degree 22
Conductor 637637
Sign −0.268+0.963i-0.268 + 0.963i
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  + 1.55·2-s + (−0.244 − 0.423i)3-s + 0.417·4-s + (−0.595 − 1.03i)5-s + (−0.380 − 0.658i)6-s − 2.46·8-s + (1.38 − 2.39i)9-s + (−0.926 − 1.60i)10-s + (−1.05 − 1.83i)11-s + (−0.102 − 0.176i)12-s + (−2.86 − 2.19i)13-s + (−0.291 + 0.504i)15-s − 4.66·16-s + 0.906·17-s + (2.14 − 3.71i)18-s + (3.34 − 5.79i)19-s + ⋯
L(s)  = 1  + 1.09·2-s + (−0.141 − 0.244i)3-s + 0.208·4-s + (−0.266 − 0.461i)5-s + (−0.155 − 0.268i)6-s − 0.870·8-s + (0.460 − 0.796i)9-s + (−0.292 − 0.507i)10-s + (−0.319 − 0.552i)11-s + (−0.0294 − 0.0510i)12-s + (−0.793 − 0.608i)13-s + (−0.0752 + 0.130i)15-s − 1.16·16-s + 0.219·17-s + (0.505 − 0.876i)18-s + (0.767 − 1.32i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.268+0.963i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.268+0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.268+0.963i-0.268 + 0.963i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(165,⋅)\chi_{637} (165, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.268+0.963i)(2,\ 637,\ (\ :1/2),\ -0.268 + 0.963i)

Particular Values

L(1)L(1) ≈\approx 1.05059−1.38367i1.05059 - 1.38367i
L(12)L(\frac12) ≈\approx 1.05059−1.38367i1.05059 - 1.38367i
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+(2.86+2.19i)T 1 + (2.86 + 2.19i)T
good2 1−1.55T+2T2 1 - 1.55T + 2T^{2}
3 1+(0.244+0.423i)T+(−1.5+2.59i)T2 1 + (0.244 + 0.423i)T + (-1.5 + 2.59i)T^{2}
5 1+(0.595+1.03i)T+(−2.5+4.33i)T2 1 + (0.595 + 1.03i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.05+1.83i)T+(−5.5+9.52i)T2 1 + (1.05 + 1.83i)T + (-5.5 + 9.52i)T^{2}
17 1−0.906T+17T2 1 - 0.906T + 17T^{2}
19 1+(−3.34+5.79i)T+(−9.5−16.4i)T2 1 + (-3.34 + 5.79i)T + (-9.5 - 16.4i)T^{2}
23 1−3.59T+23T2 1 - 3.59T + 23T^{2}
29 1+(4.25−7.37i)T+(−14.5−25.1i)T2 1 + (4.25 - 7.37i)T + (-14.5 - 25.1i)T^{2}
31 1+(2.64−4.57i)T+(−15.5−26.8i)T2 1 + (2.64 - 4.57i)T + (-15.5 - 26.8i)T^{2}
37 1−4.99T+37T2 1 - 4.99T + 37T^{2}
41 1+(−0.768+1.33i)T+(−20.5−35.5i)T2 1 + (-0.768 + 1.33i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.71+4.70i)T+(−21.5+37.2i)T2 1 + (2.71 + 4.70i)T + (-21.5 + 37.2i)T^{2}
47 1+(1.59+2.75i)T+(−23.5+40.7i)T2 1 + (1.59 + 2.75i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.41+2.44i)T+(−26.5−45.8i)T2 1 + (-1.41 + 2.44i)T + (-26.5 - 45.8i)T^{2}
59 1−10.2T+59T2 1 - 10.2T + 59T^{2}
61 1+(4.13−7.16i)T+(−30.5−52.8i)T2 1 + (4.13 - 7.16i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.87−3.24i)T+(−33.5+58.0i)T2 1 + (-1.87 - 3.24i)T + (-33.5 + 58.0i)T^{2}
71 1+(−1.26−2.19i)T+(−35.5+61.4i)T2 1 + (-1.26 - 2.19i)T + (-35.5 + 61.4i)T^{2}
73 1+(2.86−4.96i)T+(−36.5−63.2i)T2 1 + (2.86 - 4.96i)T + (-36.5 - 63.2i)T^{2}
79 1+(3.03+5.25i)T+(−39.5+68.4i)T2 1 + (3.03 + 5.25i)T + (-39.5 + 68.4i)T^{2}
83 1−11.6T+83T2 1 - 11.6T + 83T^{2}
89 1−17.7T+89T2 1 - 17.7T + 89T^{2}
97 1+(−3.10−5.37i)T+(−48.5+84.0i)T2 1 + (-3.10 - 5.37i)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

−10.46486232191646506085354736734, −9.299045005104852698129239204665, −8.730845689736413346798672044120, −7.40500887817173082708220923635, −6.64391923467102880203191589847, −5.39645820829989256143748470615, −4.94774256411562402689407929659, −3.72153108531731210608303627082, −2.83753304758176781872754146496, −0.67884128888582850562125857081, 2.19552020756986417624906489263, 3.45383237494748981207863400597, 4.41442896698131473722950264766, 5.14063902093025445240801740250, 6.06752245710807649410161224535, 7.28220558239053534132761585925, 7.88465878733558393149803126249, 9.431419963703301765679049554934, 9.907021761462274089912152266783, 11.08922256377407350331294049988

Graph of the ZZ-function along the critical line