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Properties

Label 2-507-13.4-c1-0-3
Degree 22
Conductor 507507
Sign −0.996+0.0841i-0.996 + 0.0841i
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  + (1.07 + 0.623i)2-s + (−0.5 + 0.866i)3-s + (−0.222 − 0.385i)4-s + 2.80i·5-s + (−1.07 + 0.623i)6-s + (−4.15 + 2.40i)7-s − 3.04i·8-s + (−0.499 − 0.866i)9-s + (−1.74 + 3.02i)10-s + (−1.27 − 0.733i)11-s + 0.445·12-s − 5.98·14-s + (−2.42 − 1.40i)15-s + (1.45 − 2.52i)16-s + (−1.22 − 2.11i)17-s − 1.24i·18-s + ⋯
L(s)  = 1  + (0.763 + 0.440i)2-s + (−0.288 + 0.499i)3-s + (−0.111 − 0.192i)4-s + 1.25i·5-s + (−0.440 + 0.254i)6-s + (−1.57 + 0.907i)7-s − 1.07i·8-s + (−0.166 − 0.288i)9-s + (−0.552 + 0.956i)10-s + (−0.383 − 0.221i)11-s + 0.128·12-s − 1.60·14-s + (−0.626 − 0.361i)15-s + (0.363 − 0.630i)16-s + (−0.296 − 0.513i)17-s − 0.293i·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.996+0.0841i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.996+0.0841i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.996+0.0841i-0.996 + 0.0841i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(316,⋅)\chi_{507} (316, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.996+0.0841i)(2,\ 507,\ (\ :1/2),\ -0.996 + 0.0841i)

Particular Values

L(1)L(1) ≈\approx 0.0341271−0.809995i0.0341271 - 0.809995i
L(12)L(\frac12) ≈\approx 0.0341271−0.809995i0.0341271 - 0.809995i
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+(−1.07−0.623i)T+(1+1.73i)T2 1 + (-1.07 - 0.623i)T + (1 + 1.73i)T^{2}
5 1−2.80iT−5T2 1 - 2.80iT - 5T^{2}
7 1+(4.15−2.40i)T+(3.5−6.06i)T2 1 + (4.15 - 2.40i)T + (3.5 - 6.06i)T^{2}
11 1+(1.27+0.733i)T+(5.5+9.52i)T2 1 + (1.27 + 0.733i)T + (5.5 + 9.52i)T^{2}
17 1+(1.22+2.11i)T+(−8.5+14.7i)T2 1 + (1.22 + 2.11i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.20−1.27i)T+(9.5−16.4i)T2 1 + (2.20 - 1.27i)T + (9.5 - 16.4i)T^{2}
23 1+(1.75−3.04i)T+(−11.5−19.9i)T2 1 + (1.75 - 3.04i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.925−1.60i)T+(−14.5−25.1i)T2 1 + (0.925 - 1.60i)T + (-14.5 - 25.1i)T^{2}
31 1−7.63iT−31T2 1 - 7.63iT - 31T^{2}
37 1+(−3.94−2.27i)T+(18.5+32.0i)T2 1 + (-3.94 - 2.27i)T + (18.5 + 32.0i)T^{2}
41 1+(−1.07−0.623i)T+(20.5+35.5i)T2 1 + (-1.07 - 0.623i)T + (20.5 + 35.5i)T^{2}
43 1+(−1.19−2.06i)T+(−21.5+37.2i)T2 1 + (-1.19 - 2.06i)T + (-21.5 + 37.2i)T^{2}
47 1−12.8iT−47T2 1 - 12.8iT - 47T^{2}
53 1+8.85T+53T2 1 + 8.85T + 53T^{2}
59 1+(−1.88+1.08i)T+(29.5−51.0i)T2 1 + (-1.88 + 1.08i)T + (29.5 - 51.0i)T^{2}
61 1+(−3.91−6.78i)T+(−30.5+52.8i)T2 1 + (-3.91 - 6.78i)T + (-30.5 + 52.8i)T^{2}
67 1+(3.10+1.79i)T+(33.5+58.0i)T2 1 + (3.10 + 1.79i)T + (33.5 + 58.0i)T^{2}
71 1+(−7.65+4.41i)T+(35.5−61.4i)T2 1 + (-7.65 + 4.41i)T + (35.5 - 61.4i)T^{2}
73 1+7.69iT−73T2 1 + 7.69iT - 73T^{2}
79 1+4.02T+79T2 1 + 4.02T + 79T^{2}
83 1−0.652iT−83T2 1 - 0.652iT - 83T^{2}
89 1+(5.45+3.14i)T+(44.5+77.0i)T2 1 + (5.45 + 3.14i)T + (44.5 + 77.0i)T^{2}
97 1+(−8.68+5.01i)T+(48.5−84.0i)T2 1 + (-8.68 + 5.01i)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

−11.28681881026712450336213328027, −10.36799811216535513792756046320, −9.752298723013442265836713988262, −8.972036189036442456646302633904, −7.31812354288205105141540348057, −6.30663280108642489957909219143, −6.07136252398558104498640416294, −4.89482941618729772251283644567, −3.53272781393754365451363220863, −2.85119940877162239821246398307, 0.37034449921119987946255438498, 2.33599960684082878512451041543, 3.77056661561352096860819919455, 4.49168599406759350860458093938, 5.63195061049339482550334605027, 6.60817563018595204628830174620, 7.75623168263010866564369160035, 8.641558596714122362357219655233, 9.620806532350325489799469085561, 10.59729869552078908318986216066

Graph of the ZZ-function along the critical line