Properties

Label 2-507-13.9-c1-0-19
Degree $2$
Conductor $507$
Sign $0.0128 + 0.999i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 1.5i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.866 + 1.5i)6-s + (−1.73 − 3i)7-s − 1.73·8-s + (−0.499 − 0.866i)9-s + (−1.73 + 3i)11-s − 12-s + 6·14-s + (2.49 − 4.33i)16-s + (−3 − 5.19i)17-s + 1.73·18-s + (−1.73 − 3i)19-s − 3.46·21-s + (−3 − 5.19i)22-s + ⋯
L(s)  = 1  + (−0.612 + 1.06i)2-s + (0.288 − 0.499i)3-s + (−0.250 − 0.433i)4-s + (0.353 + 0.612i)6-s + (−0.654 − 1.13i)7-s − 0.612·8-s + (−0.166 − 0.288i)9-s + (−0.522 + 0.904i)11-s − 0.288·12-s + 1.60·14-s + (0.624 − 1.08i)16-s + (−0.727 − 1.26i)17-s + 0.408·18-s + (−0.397 − 0.688i)19-s − 0.755·21-s + (−0.639 − 1.10i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290968 - 0.287260i\)
\(L(\frac12)\) \(\approx\) \(0.290968 - 0.287260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + (1.73 + 3i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.73 - 3i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.73 + 3i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + (-3.46 + 6i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.46 + 6i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-5.19 - 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.19 + 9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.73 + 3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + (3.46 - 6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.92 - 12i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.43263562530561293191616817986, −9.473024598556610034544765491971, −8.865447672719745198249203519295, −7.54680014348575957132171804626, −7.27017958399500349143551862434, −6.57274609028942619329965000870, −5.30381557664612946007544395456, −3.88752404187981442698898770881, −2.47213258723872095346986646651, −0.26348764836235493063783754692, 2.00196344104988349130777804531, 2.95834841522328463941104787985, 3.96586037906396517021802940317, 5.71132870315572181055302443724, 6.21434428146745364696109689731, 8.175185999031981602091841397726, 8.618032925928548847544400093122, 9.613084657114825132540688034066, 10.09730016357816042317696211800, 11.12552305658876342010011729395

Graph of the $Z$-function along the critical line