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Properties

Label 2-507-13.3-c1-0-8
Degree 22
Conductor 507507
Sign −0.997+0.0743i-0.997 + 0.0743i
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  + (1.34 + 2.33i)2-s + (0.5 + 0.866i)3-s + (−2.62 + 4.54i)4-s + 1.04·5-s + (−1.34 + 2.33i)6-s + (0.277 − 0.480i)7-s − 8.74·8-s + (−0.499 + 0.866i)9-s + (1.41 + 2.44i)10-s + (1.45 + 2.52i)11-s − 5.24·12-s + 1.49·14-s + (0.524 + 0.908i)15-s + (−6.51 − 11.2i)16-s + (2.42 − 4.20i)17-s − 2.69·18-s + ⋯
L(s)  = 1  + (0.951 + 1.64i)2-s + (0.288 + 0.499i)3-s + (−1.31 + 2.27i)4-s + 0.469·5-s + (−0.549 + 0.951i)6-s + (0.104 − 0.181i)7-s − 3.09·8-s + (−0.166 + 0.288i)9-s + (0.446 + 0.773i)10-s + (0.438 + 0.760i)11-s − 1.51·12-s + 0.399·14-s + (0.135 + 0.234i)15-s + (−1.62 − 2.82i)16-s + (0.588 − 1.01i)17-s − 0.634·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.997+0.0743i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0743i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.997+0.0743i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.997+0.0743i-0.997 + 0.0743i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(484,⋅)\chi_{507} (484, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.997+0.0743i)(2,\ 507,\ (\ :1/2),\ -0.997 + 0.0743i)

Particular Values

L(1)L(1) ≈\approx 0.0906094−2.43525i0.0906094 - 2.43525i
L(12)L(\frac12) ≈\approx 0.0906094−2.43525i0.0906094 - 2.43525i
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+(−1.34−2.33i)T+(−1+1.73i)T2 1 + (-1.34 - 2.33i)T + (-1 + 1.73i)T^{2}
5 1−1.04T+5T2 1 - 1.04T + 5T^{2}
7 1+(−0.277+0.480i)T+(−3.5−6.06i)T2 1 + (-0.277 + 0.480i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1.45−2.52i)T+(−5.5+9.52i)T2 1 + (-1.45 - 2.52i)T + (-5.5 + 9.52i)T^{2}
17 1+(−2.42+4.20i)T+(−8.5−14.7i)T2 1 + (-2.42 + 4.20i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.376−0.652i)T+(−9.5−16.4i)T2 1 + (0.376 - 0.652i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.88+4.99i)T+(−11.5+19.9i)T2 1 + (2.88 + 4.99i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.955−1.65i)T+(−14.5+25.1i)T2 1 + (-0.955 - 1.65i)T + (-14.5 + 25.1i)T^{2}
31 1−9.51T+31T2 1 - 9.51T + 31T^{2}
37 1+(−2.87−4.98i)T+(−18.5+32.0i)T2 1 + (-2.87 - 4.98i)T + (-18.5 + 32.0i)T^{2}
41 1+(−2.45−4.25i)T+(−20.5+35.5i)T2 1 + (-2.45 - 4.25i)T + (-20.5 + 35.5i)T^{2}
43 1+(−5.54+9.61i)T+(−21.5−37.2i)T2 1 + (-5.54 + 9.61i)T + (-21.5 - 37.2i)T^{2}
47 1+0.753T+47T2 1 + 0.753T + 47T^{2}
53 1+7.58T+53T2 1 + 7.58T + 53T^{2}
59 1+(−2.04+3.54i)T+(−29.5−51.0i)T2 1 + (-2.04 + 3.54i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.71+2.96i)T+(−30.5−52.8i)T2 1 + (-1.71 + 2.96i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.936+1.62i)T+(−33.5+58.0i)T2 1 + (0.936 + 1.62i)T + (-33.5 + 58.0i)T^{2}
71 1+(5.25−9.09i)T+(−35.5−61.4i)T2 1 + (5.25 - 9.09i)T + (-35.5 - 61.4i)T^{2}
73 1+10.4T+73T2 1 + 10.4T + 73T^{2}
79 1−1.33T+79T2 1 - 1.33T + 79T^{2}
83 1+2.64T+83T2 1 + 2.64T + 83T^{2}
89 1+(−4.96−8.59i)T+(−44.5+77.0i)T2 1 + (-4.96 - 8.59i)T + (-44.5 + 77.0i)T^{2}
97 1+(−8.53+14.7i)T+(−48.5−84.0i)T2 1 + (-8.53 + 14.7i)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

−11.74676510240874458923900059027, −10.12868953291045272046734106401, −9.382190512647140656892717002909, −8.379451858511615424106482198443, −7.59940198985359772739241996957, −6.66082296115319693703564757596, −5.82622542458376858618406577847, −4.77135688358494427192761258073, −4.15746545763124965180406677480, −2.81355308710825158780751388134, 1.18866325473658283854390771567, 2.27421361911415527655537154334, 3.36801796273384047003738867504, 4.31191163403381028596272759754, 5.73266012847901523304275944795, 6.15071238785591016747396070070, 7.969382278860365385591019857795, 9.061397919949978956209857694465, 9.831743737037203823011563929265, 10.64165252177549007667040555709

Graph of the ZZ-function along the critical line