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Properties

Label 2-637-7.2-c1-0-24
Degree 22
Conductor 637637
Sign 0.900−0.435i0.900 - 0.435i
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  + (0.588 − 1.01i)2-s + (1.67 + 2.90i)3-s + (0.308 + 0.533i)4-s + (1.57 − 2.72i)5-s + 3.94·6-s + 3.07·8-s + (−4.11 + 7.12i)9-s + (−1.85 − 3.20i)10-s + (0.386 + 0.669i)11-s + (−1.03 + 1.78i)12-s − 13-s + 10.5·15-s + (1.19 − 2.06i)16-s + (−2.87 − 4.98i)17-s + (4.83 + 8.38i)18-s + (−0.611 + 1.05i)19-s + ⋯
L(s)  = 1  + (0.415 − 0.720i)2-s + (0.967 + 1.67i)3-s + (0.154 + 0.266i)4-s + (0.704 − 1.21i)5-s + 1.60·6-s + 1.08·8-s + (−1.37 + 2.37i)9-s + (−0.585 − 1.01i)10-s + (0.116 + 0.201i)11-s + (−0.298 + 0.516i)12-s − 0.277·13-s + 2.72·15-s + (0.298 − 0.516i)16-s + (−0.697 − 1.20i)17-s + (1.14 + 1.97i)18-s + (−0.140 + 0.242i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.900−0.435i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.900−0.435i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.900−0.435i0.900 - 0.435i
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), 0.900−0.435i)(2,\ 637,\ (\ :1/2),\ 0.900 - 0.435i)

Particular Values

L(1)L(1) ≈\approx 2.88270+0.661254i2.88270 + 0.661254i
L(12)L(\frac12) ≈\approx 2.88270+0.661254i2.88270 + 0.661254i
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+(−0.588+1.01i)T+(−1−1.73i)T2 1 + (-0.588 + 1.01i)T + (-1 - 1.73i)T^{2}
3 1+(−1.67−2.90i)T+(−1.5+2.59i)T2 1 + (-1.67 - 2.90i)T + (-1.5 + 2.59i)T^{2}
5 1+(−1.57+2.72i)T+(−2.5−4.33i)T2 1 + (-1.57 + 2.72i)T + (-2.5 - 4.33i)T^{2}
11 1+(−0.386−0.669i)T+(−5.5+9.52i)T2 1 + (-0.386 - 0.669i)T + (-5.5 + 9.52i)T^{2}
17 1+(2.87+4.98i)T+(−8.5+14.7i)T2 1 + (2.87 + 4.98i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.611−1.05i)T+(−9.5−16.4i)T2 1 + (0.611 - 1.05i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.49+2.59i)T+(−11.5−19.9i)T2 1 + (-1.49 + 2.59i)T + (-11.5 - 19.9i)T^{2}
29 1+2.46T+29T2 1 + 2.46T + 29T^{2}
31 1+(3.06+5.31i)T+(−15.5+26.8i)T2 1 + (3.06 + 5.31i)T + (-15.5 + 26.8i)T^{2}
37 1+(2.49−4.32i)T+(−18.5−32.0i)T2 1 + (2.49 - 4.32i)T + (-18.5 - 32.0i)T^{2}
41 1−2.55T+41T2 1 - 2.55T + 41T^{2}
43 1+2.73T+43T2 1 + 2.73T + 43T^{2}
47 1+(−2.68+4.65i)T+(−23.5−40.7i)T2 1 + (-2.68 + 4.65i)T + (-23.5 - 40.7i)T^{2}
53 1+(−4.89−8.47i)T+(−26.5+45.8i)T2 1 + (-4.89 - 8.47i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1.25−2.16i)T+(−29.5+51.0i)T2 1 + (-1.25 - 2.16i)T + (-29.5 + 51.0i)T^{2}
61 1+(5.45−9.45i)T+(−30.5−52.8i)T2 1 + (5.45 - 9.45i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.16+3.74i)T+(−33.5+58.0i)T2 1 + (2.16 + 3.74i)T + (-33.5 + 58.0i)T^{2}
71 1−10.6T+71T2 1 - 10.6T + 71T^{2}
73 1+(2.58+4.48i)T+(−36.5+63.2i)T2 1 + (2.58 + 4.48i)T + (-36.5 + 63.2i)T^{2}
79 1+(−0.271+0.469i)T+(−39.5−68.4i)T2 1 + (-0.271 + 0.469i)T + (-39.5 - 68.4i)T^{2}
83 1+15.2T+83T2 1 + 15.2T + 83T^{2}
89 1+(−4.61+7.99i)T+(−44.5−77.0i)T2 1 + (-4.61 + 7.99i)T + (-44.5 - 77.0i)T^{2}
97 1−1.26T+97T2 1 - 1.26T + 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

−10.52497014344781328831634305083, −9.744017161548606273303197717438, −9.109361170913968095149314092274, −8.468389069415162227453078279944, −7.38799817246054000915243844587, −5.49538150347177319738532223989, −4.67743245249855001822238429271, −4.16148743751562304029694319533, −2.94770265265342989769607153119, −2.05415474336061865201920730023, 1.63388226194346806761296036425, 2.41639395086857898846195258357, 3.62247187725756920003582783014, 5.57650120181352931698054555442, 6.39731184263339212083275989287, 6.85144515537506227261058178051, 7.50466951730622590299588007797, 8.458846313270456170985010642466, 9.465149963844054735312610902379, 10.61115729617675603293422268739

Graph of the ZZ-function along the critical line