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Properties

Label 2-507-13.10-c1-0-4
Degree 22
Conductor 507507
Sign −0.944+0.327i-0.944 + 0.327i
Analytic cond. 4.048414.04841
Root an. cond. 2.012062.01206
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  + (−1.77 + 1.02i)2-s + (0.5 + 0.866i)3-s + (1.09 − 1.90i)4-s + 3.35i·5-s + (−1.77 − 1.02i)6-s + (1.94 + 1.12i)7-s + 0.405i·8-s + (−0.499 + 0.866i)9-s + (−3.43 − 5.95i)10-s + (−4.27 + 2.46i)11-s + 2.19·12-s − 4.60·14-s + (−2.90 + 1.67i)15-s + (1.78 + 3.08i)16-s + (0.455 − 0.789i)17-s − 2.04i·18-s + ⋯
L(s)  = 1  + (−1.25 + 0.724i)2-s + (0.288 + 0.499i)3-s + (0.549 − 0.951i)4-s + 1.50i·5-s + (−0.724 − 0.418i)6-s + (0.735 + 0.424i)7-s + 0.143i·8-s + (−0.166 + 0.288i)9-s + (−1.08 − 1.88i)10-s + (−1.28 + 0.744i)11-s + 0.634·12-s − 1.23·14-s + (−0.750 + 0.433i)15-s + (0.445 + 0.771i)16-s + (0.110 − 0.191i)17-s − 0.482i·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.944+0.327i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.944+0.327i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.944+0.327i-0.944 + 0.327i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(361,⋅)\chi_{507} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.944+0.327i)(2,\ 507,\ (\ :1/2),\ -0.944 + 0.327i)

Particular Values

L(1)L(1) ≈\approx 0.114952−0.681747i0.114952 - 0.681747i
L(12)L(\frac12) ≈\approx 0.114952−0.681747i0.114952 - 0.681747i
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1 1
good2 1+(1.77−1.02i)T+(1−1.73i)T2 1 + (1.77 - 1.02i)T + (1 - 1.73i)T^{2}
5 1−3.35iT−5T2 1 - 3.35iT - 5T^{2}
7 1+(−1.94−1.12i)T+(3.5+6.06i)T2 1 + (-1.94 - 1.12i)T + (3.5 + 6.06i)T^{2}
11 1+(4.27−2.46i)T+(5.5−9.52i)T2 1 + (4.27 - 2.46i)T + (5.5 - 9.52i)T^{2}
17 1+(−0.455+0.789i)T+(−8.5−14.7i)T2 1 + (-0.455 + 0.789i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.29−1.90i)T+(9.5+16.4i)T2 1 + (-3.29 - 1.90i)T + (9.5 + 16.4i)T^{2}
23 1+(−1.01−1.75i)T+(−11.5+19.9i)T2 1 + (-1.01 - 1.75i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.96−3.41i)T+(−14.5+25.1i)T2 1 + (-1.96 - 3.41i)T + (-14.5 + 25.1i)T^{2}
31 1+8.82iT−31T2 1 + 8.82iT - 31T^{2}
37 1+(7.62−4.40i)T+(18.5−32.0i)T2 1 + (7.62 - 4.40i)T + (18.5 - 32.0i)T^{2}
41 1+(−6.00+3.46i)T+(20.5−35.5i)T2 1 + (-6.00 + 3.46i)T + (20.5 - 35.5i)T^{2}
43 1+(1.14−1.97i)T+(−21.5−37.2i)T2 1 + (1.14 - 1.97i)T + (-21.5 - 37.2i)T^{2}
47 1+3.80iT−47T2 1 + 3.80iT - 47T^{2}
53 1−0.542T+53T2 1 - 0.542T + 53T^{2}
59 1+(4.08+2.35i)T+(29.5+51.0i)T2 1 + (4.08 + 2.35i)T + (29.5 + 51.0i)T^{2}
61 1+(1.83−3.18i)T+(−30.5−52.8i)T2 1 + (1.83 - 3.18i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.31−0.760i)T+(33.5−58.0i)T2 1 + (1.31 - 0.760i)T + (33.5 - 58.0i)T^{2}
71 1+(2.05+1.18i)T+(35.5+61.4i)T2 1 + (2.05 + 1.18i)T + (35.5 + 61.4i)T^{2}
73 1+7.41iT−73T2 1 + 7.41iT - 73T^{2}
79 1+3.74T+79T2 1 + 3.74T + 79T^{2}
83 1−2.30iT−83T2 1 - 2.30iT - 83T^{2}
89 1+(−8.71+5.02i)T+(44.5−77.0i)T2 1 + (-8.71 + 5.02i)T + (44.5 - 77.0i)T^{2}
97 1+(−13.9−8.06i)T+(48.5+84.0i)T2 1 + (-13.9 - 8.06i)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

−10.89848054492272241118139620295, −10.27928292061870484997570873945, −9.696228988541895797044014778544, −8.646724201429354170525290449164, −7.63428222962658026842625426826, −7.38551782863934856907464156999, −6.13685833940453668644980781511, −5.03157317547601195081505123986, −3.39257626340778997560335807953, −2.16748650816633686982198396665, 0.61778297584089204469821026525, 1.58961213543626588221267076631, 2.96641589143885579295620926725, 4.73734372272029192425151700777, 5.56192139015211603365997862946, 7.37274206208676278693255798461, 8.140869535350351505225197968980, 8.591115265999338478399146728125, 9.353219369904167319192675457828, 10.42731170274068395992734656646

Graph of the ZZ-function along the critical line