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Properties

Label 2-63-63.20-c1-0-4
Degree 22
Conductor 6363
Sign −0.692+0.721i-0.692 + 0.721i
Analytic cond. 0.5030570.503057
Root an. cond. 0.7092650.709265
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  + (−1.97 − 1.13i)2-s + (0.578 − 1.63i)3-s + (1.59 + 2.75i)4-s + (−0.717 − 1.24i)5-s + (−2.99 + 2.55i)6-s + (−2.40 − 1.11i)7-s − 2.69i·8-s + (−2.33 − 1.88i)9-s + 3.26i·10-s + (2.80 + 1.61i)11-s + (5.41 − 1.00i)12-s + (4.43 − 2.55i)13-s + (3.47 + 4.92i)14-s + (−2.44 + 0.451i)15-s + (0.119 − 0.207i)16-s + 1.09·17-s + ⋯
L(s)  = 1  + (−1.39 − 0.804i)2-s + (0.334 − 0.942i)3-s + (0.795 + 1.37i)4-s + (−0.320 − 0.555i)5-s + (−1.22 + 1.04i)6-s + (−0.907 − 0.419i)7-s − 0.951i·8-s + (−0.776 − 0.629i)9-s + 1.03i·10-s + (0.844 + 0.487i)11-s + (1.56 − 0.289i)12-s + (1.22 − 0.709i)13-s + (0.927 + 1.31i)14-s + (−0.630 + 0.116i)15-s + (0.0298 − 0.0517i)16-s + 0.264·17-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((−0.692+0.721i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+1/2) L(s)=((−0.692+0.721i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: −0.692+0.721i-0.692 + 0.721i
Analytic conductor: 0.5030570.503057
Root analytic conductor: 0.7092650.709265
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ63(20,⋅)\chi_{63} (20, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :1/2), −0.692+0.721i)(2,\ 63,\ (\ :1/2),\ -0.692 + 0.721i)

Particular Values

L(1)L(1) ≈\approx 0.186541−0.437924i0.186541 - 0.437924i
L(12)L(\frac12) ≈\approx 0.186541−0.437924i0.186541 - 0.437924i
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)
bad3 1+(−0.578+1.63i)T 1 + (-0.578 + 1.63i)T
7 1+(2.40+1.11i)T 1 + (2.40 + 1.11i)T
good2 1+(1.97+1.13i)T+(1+1.73i)T2 1 + (1.97 + 1.13i)T + (1 + 1.73i)T^{2}
5 1+(0.717+1.24i)T+(−2.5+4.33i)T2 1 + (0.717 + 1.24i)T + (-2.5 + 4.33i)T^{2}
11 1+(−2.80−1.61i)T+(5.5+9.52i)T2 1 + (-2.80 - 1.61i)T + (5.5 + 9.52i)T^{2}
13 1+(−4.43+2.55i)T+(6.5−11.2i)T2 1 + (-4.43 + 2.55i)T + (6.5 - 11.2i)T^{2}
17 1−1.09T+17T2 1 - 1.09T + 17T^{2}
19 1−4.48iT−19T2 1 - 4.48iT - 19T^{2}
23 1+(−3.47+2.00i)T+(11.5−19.9i)T2 1 + (-3.47 + 2.00i)T + (11.5 - 19.9i)T^{2}
29 1+(−1.02−0.593i)T+(14.5+25.1i)T2 1 + (-1.02 - 0.593i)T + (14.5 + 25.1i)T^{2}
31 1+(3.24−1.87i)T+(15.5−26.8i)T2 1 + (3.24 - 1.87i)T + (15.5 - 26.8i)T^{2}
37 1+0.239T+37T2 1 + 0.239T + 37T^{2}
41 1+(−3.71−6.43i)T+(−20.5+35.5i)T2 1 + (-3.71 - 6.43i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.82−6.62i)T+(−21.5−37.2i)T2 1 + (3.82 - 6.62i)T + (-21.5 - 37.2i)T^{2}
47 1+(2.11−3.65i)T+(−23.5−40.7i)T2 1 + (2.11 - 3.65i)T + (-23.5 - 40.7i)T^{2}
53 1+7.01iT−53T2 1 + 7.01iT - 53T^{2}
59 1+(4.73+8.20i)T+(−29.5+51.0i)T2 1 + (4.73 + 8.20i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.82+1.63i)T+(30.5+52.8i)T2 1 + (2.82 + 1.63i)T + (30.5 + 52.8i)T^{2}
67 1+(0.330+0.571i)T+(−33.5+58.0i)T2 1 + (0.330 + 0.571i)T + (-33.5 + 58.0i)T^{2}
71 1+3.82iT−71T2 1 + 3.82iT - 71T^{2}
73 1−7.31iT−73T2 1 - 7.31iT - 73T^{2}
79 1+(1.83−3.16i)T+(−39.5−68.4i)T2 1 + (1.83 - 3.16i)T + (-39.5 - 68.4i)T^{2}
83 1+(5.45−9.44i)T+(−41.5−71.8i)T2 1 + (5.45 - 9.44i)T + (-41.5 - 71.8i)T^{2}
89 1−13.6T+89T2 1 - 13.6T + 89T^{2}
97 1+(−2.69−1.55i)T+(48.5+84.0i)T2 1 + (-2.69 - 1.55i)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

−14.43774371613803619824094158064, −12.94297198642855350628004629658, −12.31681668555976572621189403781, −11.09589945699888816021294856616, −9.788409594005608981836469380657, −8.740360628311118463668635822796, −7.85615946311581143094563245081, −6.46436863352392689320904694421, −3.34091255397121325083077074527, −1.16972861696127013472767891170, 3.53786021103306902518950838853, 6.00566203157387147764646895058, 7.16224305504598902343677629083, 8.894644153988514565837566317656, 9.117169602911260789621143059951, 10.50808529411448498470970372816, 11.43418172654709211579379068554, 13.57302218270449218147251419517, 14.91724853030596000616581729866, 15.64183342901212976672163941506

Graph of the ZZ-function along the critical line