Properties

Label 2-507-13.9-c1-0-14
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.207 + 0.358i)2-s + (−0.5 + 0.866i)3-s + (0.914 + 1.58i)4-s − 2.82·5-s + (−0.207 − 0.358i)6-s + (−1.41 − 2.44i)7-s − 1.58·8-s + (−0.499 − 0.866i)9-s + (0.585 − 1.01i)10-s + (1 − 1.73i)11-s − 1.82·12-s + 1.17·14-s + (1.41 − 2.44i)15-s + (−1.49 + 2.59i)16-s + (−3.82 − 6.63i)17-s + 0.414·18-s + ⋯
L(s)  = 1  + (−0.146 + 0.253i)2-s + (−0.288 + 0.499i)3-s + (0.457 + 0.791i)4-s − 1.26·5-s + (−0.0845 − 0.146i)6-s + (−0.534 − 0.925i)7-s − 0.560·8-s + (−0.166 − 0.288i)9-s + (0.185 − 0.320i)10-s + (0.301 − 0.522i)11-s − 0.527·12-s + 0.313·14-s + (0.365 − 0.632i)15-s + (−0.374 + 0.649i)16-s + (−0.928 − 1.60i)17-s + 0.0976·18-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

\(L(1)\) \(\approx\) \(0.238097 - 0.235063i\)
\(L(\frac12)\) \(\approx\) \(0.238097 - 0.235063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.207 - 0.358i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 + (1.41 + 2.44i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.82 + 6.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 + (-3.82 + 6.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.58 - 4.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.828 - 1.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (3.82 + 6.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.65 + 11.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.41 - 5.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1 + 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.343T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 3.65T + 83T^{2} \)
89 \( 1 + (7.41 - 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.82 + 3.16i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.14772970363072103721666723090, −9.746676257126419763482239369480, −8.812003771421564371270107471506, −7.84922139129924135166133311769, −7.12504765337616079950131112950, −6.38003513046821651779530456428, −4.72640259683560524811398165544, −3.79536080434194564220302980263, −3.05442107698060846600322638950, −0.20936790227373933743897108075, 1.68544522410388219566703565469, 3.03746318972948242340605674710, 4.45577918369397104734380978904, 5.74418499667077562132288631447, 6.53482904977824713737872649975, 7.39463542524467869950650409295, 8.510387771172450664949532518770, 9.345120276430629891300716868592, 10.42007397234207754512174234294, 11.35255559150932765050858955164

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