Properties

Label 2-637-91.9-c1-0-4
Degree 22
Conductor 637637
Sign 0.9100.413i0.910 - 0.413i
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 + 0.753·3-s + (−1.95 + 3.39i)4-s + (−0.170 + 0.295i)5-s + (−0.916 − 1.58i)6-s + 4.65·8-s − 2.43·9-s + 0.830·10-s − 2.43·11-s + (−1.47 + 2.55i)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 + (2.95 + 5.12i)18-s − 6.29·19-s + ⋯
L(s)  = 1  + (−0.859 − 1.48i)2-s + 0.435·3-s + (−0.978 + 1.69i)4-s + (−0.0763 + 0.132i)5-s + (−0.374 − 0.647i)6-s + 1.64·8-s − 0.810·9-s + 0.262·10-s − 0.733·11-s + (−0.425 + 0.737i)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 + (0.697 + 1.20i)18-s − 1.44·19-s + ⋯

Functional equation

Λ(s)=(637s/2ΓC(s)L(s)=((0.9100.413i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2ΓC(s+1/2)L(s)=((0.9100.413i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72137^{2} \cdot 13
Sign: 0.9100.413i0.910 - 0.413i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(373,)\chi_{637} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.9100.413i)(2,\ 637,\ (\ :1/2),\ 0.910 - 0.413i)

Particular Values

L(1)L(1) \approx 0.518830+0.112174i0.518830 + 0.112174i
L(12)L(\frac12) \approx 0.518830+0.112174i0.518830 + 0.112174i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
13 1+(2.502.59i)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 10.753T+3T2 1 - 0.753T + 3T^{2}
5 1+(0.1700.295i)T+(2.54.33i)T2 1 + (0.170 - 0.295i)T + (-2.5 - 4.33i)T^{2}
11 1+2.43T+11T2 1 + 2.43T + 11T^{2}
17 1+(0.974+1.68i)T+(8.514.7i)T2 1 + (-0.974 + 1.68i)T + (-8.5 - 14.7i)T^{2}
19 1+6.29T+19T2 1 + 6.29T + 19T^{2}
23 1+(1.843.19i)T+(11.5+19.9i)T2 1 + (-1.84 - 3.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.223.84i)T+(14.525.1i)T2 1 + (2.22 - 3.84i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.9871.71i)T+(15.5+26.8i)T2 1 + (-0.987 - 1.71i)T + (-15.5 + 26.8i)T^{2}
37 1+(4.818.33i)T+(18.5+32.0i)T2 1 + (-4.81 - 8.33i)T + (-18.5 + 32.0i)T^{2}
41 1+(6.2610.8i)T+(20.535.5i)T2 1 + (6.26 - 10.8i)T + (-20.5 - 35.5i)T^{2}
43 1+(4.207.28i)T+(21.5+37.2i)T2 1 + (-4.20 - 7.28i)T + (-21.5 + 37.2i)T^{2}
47 1+(4.50+7.79i)T+(23.540.7i)T2 1 + (-4.50 + 7.79i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.746+1.29i)T+(26.5+45.8i)T2 1 + (0.746 + 1.29i)T + (-26.5 + 45.8i)T^{2}
59 1+(0.3130.542i)T+(29.551.0i)T2 1 + (0.313 - 0.542i)T + (-29.5 - 51.0i)T^{2}
61 1+1.14T+61T2 1 + 1.14T + 61T^{2}
67 1+5.59T+67T2 1 + 5.59T + 67T^{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+(5.95+10.3i)T+(36.5+63.2i)T2 1 + (5.95 + 10.3i)T + (-36.5 + 63.2i)T^{2}
79 1+(2.233.87i)T+(39.568.4i)T2 1 + (2.23 - 3.87i)T + (-39.5 - 68.4i)T^{2}
83 11.41T+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(1αj,pps)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

−10.74156196655733833986584841084, −9.820825683583443916993675815681, −8.957123227238904047699948640436, −8.481187572506083493844367524053, −7.57628010844268431990553783704, −6.21670511005601086260079659492, −4.74276953406743188170809781075, −3.43955744881771086119678571796, −2.76111739508765026665203052232, −1.53643566161148542785053123402, 0.36453354893281442051376943767, 2.54397241791588699981131003131, 4.17258647250796809360704273054, 5.57019992730040490130479764472, 6.03392700918412714462102576527, 7.16744331647070007449129894764, 8.111106345820478502727110745413, 8.501911411568756452069907564365, 9.176581945630727948690227827998, 10.38706769655740612920630871283

Graph of the ZZ-function along the critical line