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Properties

Label 2-507-13.12-c1-0-11
Degree $2$
Conductor $507$
Sign $0.246 - 0.969i$
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  + 1.24i·2-s + 3-s + 0.445·4-s − 2.80i·5-s + 1.24i·6-s + 4.80i·7-s + 3.04i·8-s + 9-s + 3.49·10-s − 1.46i·11-s + 0.445·12-s − 5.98·14-s − 2.80i·15-s − 2.91·16-s + 2.44·17-s + 1.24i·18-s + ⋯
L(s)  = 1  + 0.881i·2-s + 0.577·3-s + 0.222·4-s − 1.25i·5-s + 0.509i·6-s + 1.81i·7-s + 1.07i·8-s + 0.333·9-s + 1.10·10-s − 0.442i·11-s + 0.128·12-s − 1.60·14-s − 0.723i·15-s − 0.727·16-s + 0.593·17-s + 0.293i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57274 + 1.22231i\)
\(L(\frac12)\) \(\approx\) \(1.57274 + 1.22231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - 1.24iT - 2T^{2} \)
5 \( 1 + 2.80iT - 5T^{2} \)
7 \( 1 - 4.80iT - 7T^{2} \)
11 \( 1 + 1.46iT - 11T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 - 2.54iT - 19T^{2} \)
23 \( 1 - 3.51T + 23T^{2} \)
29 \( 1 - 1.85T + 29T^{2} \)
31 \( 1 + 7.63iT - 31T^{2} \)
37 \( 1 - 4.55iT - 37T^{2} \)
41 \( 1 - 1.24iT - 41T^{2} \)
43 \( 1 + 2.38T + 43T^{2} \)
47 \( 1 + 12.8iT - 47T^{2} \)
53 \( 1 + 8.85T + 53T^{2} \)
59 \( 1 + 2.17iT - 59T^{2} \)
61 \( 1 + 7.82T + 61T^{2} \)
67 \( 1 + 3.58iT - 67T^{2} \)
71 \( 1 + 8.83iT - 71T^{2} \)
73 \( 1 - 7.69iT - 73T^{2} \)
79 \( 1 + 4.02T + 79T^{2} \)
83 \( 1 + 0.652iT - 83T^{2} \)
89 \( 1 + 6.29iT - 89T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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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

−11.35097011052072648799434218004, −9.818510484441466866469291056515, −8.882467422157888050966019233428, −8.461409046357297124101675337925, −7.74712749259316213887650668173, −6.33951970429966534081112723030, −5.57126674527757022850408423181, −4.84836584027712463498210183411, −3.05734912950708440012946254322, −1.83456533850164448457965167917, 1.30744597933420542194274042478, 2.82728474994628151505669253728, 3.47790932776250483432896865000, 4.51797539478318427931039286220, 6.56357592278166211686071906756, 7.11572011120892861508367109617, 7.72428523563077006371524010172, 9.346158685123999019233196932076, 10.21699150053343485460242225864, 10.67968193076440914608206262444

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