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Properties

Label 2-637-13.9-c1-0-23
Degree 22
Conductor 637637
Sign 0.999+0.0251i0.999 + 0.0251i
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  + (−1.21 + 2.10i)2-s + (0.376 − 0.652i)3-s + (−1.95 − 3.39i)4-s − 0.341·5-s + (0.916 + 1.58i)6-s + 4.65·8-s + (1.21 + 2.10i)9-s + (0.415 − 0.719i)10-s + (1.21 − 2.10i)11-s − 2.95·12-s + (−2.50 − 2.59i)13-s + (−0.128 + 0.222i)15-s + (−1.74 + 3.02i)16-s + (−0.974 − 1.68i)17-s − 5.91·18-s + (−3.14 − 5.44i)19-s + ⋯
L(s)  = 1  + (−0.859 + 1.48i)2-s + (0.217 − 0.376i)3-s + (−0.978 − 1.69i)4-s − 0.152·5-s + (0.374 + 0.647i)6-s + 1.64·8-s + (0.405 + 0.702i)9-s + (0.131 − 0.227i)10-s + (0.366 − 0.635i)11-s − 0.851·12-s + (−0.693 − 0.720i)13-s + (−0.0332 + 0.0575i)15-s + (−0.437 + 0.757i)16-s + (−0.236 − 0.409i)17-s − 1.39·18-s + (−0.721 − 1.24i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.999+0.0251i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0251i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.999+0.0251i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.999+0.0251i0.999 + 0.0251i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(295,⋅)\chi_{637} (295, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.999+0.0251i)(2,\ 637,\ (\ :1/2),\ 0.999 + 0.0251i)

Particular Values

L(1)L(1) ≈\approx 0.755216−0.00948345i0.755216 - 0.00948345i
L(12)L(\frac12) ≈\approx 0.755216−0.00948345i0.755216 - 0.00948345i
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+(2.50+2.59i)T 1 + (2.50 + 2.59i)T
good2 1+(1.21−2.10i)T+(−1−1.73i)T2 1 + (1.21 - 2.10i)T + (-1 - 1.73i)T^{2}
3 1+(−0.376+0.652i)T+(−1.5−2.59i)T2 1 + (-0.376 + 0.652i)T + (-1.5 - 2.59i)T^{2}
5 1+0.341T+5T2 1 + 0.341T + 5T^{2}
11 1+(−1.21+2.10i)T+(−5.5−9.52i)T2 1 + (-1.21 + 2.10i)T + (-5.5 - 9.52i)T^{2}
17 1+(0.974+1.68i)T+(−8.5+14.7i)T2 1 + (0.974 + 1.68i)T + (-8.5 + 14.7i)T^{2}
19 1+(3.14+5.44i)T+(−9.5+16.4i)T2 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.84+3.19i)T+(−11.5−19.9i)T2 1 + (-1.84 + 3.19i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.22−3.84i)T+(−14.5−25.1i)T2 1 + (2.22 - 3.84i)T + (-14.5 - 25.1i)T^{2}
31 1−1.97T+31T2 1 - 1.97T + 31T^{2}
37 1+(−4.81+8.33i)T+(−18.5−32.0i)T2 1 + (-4.81 + 8.33i)T + (-18.5 - 32.0i)T^{2}
41 1+(−6.26+10.8i)T+(−20.5−35.5i)T2 1 + (-6.26 + 10.8i)T + (-20.5 - 35.5i)T^{2}
43 1+(−4.20−7.28i)T+(−21.5+37.2i)T2 1 + (-4.20 - 7.28i)T + (-21.5 + 37.2i)T^{2}
47 1−9.00T+47T2 1 - 9.00T + 47T^{2}
53 1−1.49T+53T2 1 - 1.49T + 53T^{2}
59 1+(−0.313−0.542i)T+(−29.5+51.0i)T2 1 + (-0.313 - 0.542i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.571+0.990i)T+(−30.5+52.8i)T2 1 + (0.571 + 0.990i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2.79+4.84i)T+(−33.5−58.0i)T2 1 + (-2.79 + 4.84i)T + (-33.5 - 58.0i)T^{2}
71 1+(4.74+8.22i)T+(−35.5+61.4i)T2 1 + (4.74 + 8.22i)T + (-35.5 + 61.4i)T^{2}
73 1+11.9T+73T2 1 + 11.9T + 73T^{2}
79 1−4.47T+79T2 1 - 4.47T + 79T^{2}
83 1+1.41T+83T2 1 + 1.41T + 83T^{2}
89 1+(−6.22+10.7i)T+(−44.5−77.0i)T2 1 + (-6.22 + 10.7i)T + (-44.5 - 77.0i)T^{2}
97 1+(−5.13−8.90i)T+(−48.5+84.0i)T2 1 + (-5.13 - 8.90i)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

−10.38795564648361009113412217962, −9.220968817348667323199288076063, −8.765335982669256402004550709848, −7.64867495792063346925017572549, −7.37170606598318398420856886119, −6.35436406310115565615875567914, −5.42259412873009241545439419779, −4.41700022553385772334277498546, −2.50453516339212674287377002269, −0.57490128015347619171821733892, 1.40864519121509738458608423867, 2.53676864066042077082901779177, 3.90277145068512753118606146916, 4.30026099293465417809690979946, 6.16414645069413888699547509016, 7.40915010236327681443229541896, 8.332052993484195798280829631030, 9.293994723252685435559612367807, 9.755471450692216214522047065176, 10.35485341885716250528492233085

Graph of the ZZ-function along the critical line