Properties

Label 2-507-13.12-c1-0-10
Degree $2$
Conductor $507$
Sign $0.832 - 0.554i$
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  − 0.414i·2-s + 3-s + 1.82·4-s + 2.82i·5-s − 0.414i·6-s + 2.82i·7-s − 1.58i·8-s + 9-s + 1.17·10-s − 2i·11-s + 1.82·12-s + 1.17·14-s + 2.82i·15-s + 3·16-s − 7.65·17-s − 0.414i·18-s + ⋯
L(s)  = 1  − 0.292i·2-s + 0.577·3-s + 0.914·4-s + 1.26i·5-s − 0.169i·6-s + 1.06i·7-s − 0.560i·8-s + 0.333·9-s + 0.370·10-s − 0.603i·11-s + 0.527·12-s + 0.313·14-s + 0.730i·15-s + 0.750·16-s − 1.85·17-s − 0.0976i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.832 - 0.554i$
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: \((2,\ 507,\ (\ :1/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98701 + 0.601620i\)
\(L(\frac12)\) \(\approx\) \(1.98701 + 0.601620i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414iT - 2T^{2} \)
5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 1.17iT - 31T^{2} \)
37 \( 1 + 7.65iT - 37T^{2} \)
41 \( 1 + 5.17iT - 41T^{2} \)
43 \( 1 - 1.65T + 43T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 7.65iT - 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + 6.82iT - 67T^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 - 0.343iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 3.65iT - 83T^{2} \)
89 \( 1 - 14.8iT - 89T^{2} \)
97 \( 1 + 3.65iT - 97T^{2} \)
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

−10.95145106171776430141403217366, −10.38928897116044569767685879308, −9.182374177958968420282998651461, −8.400504597629620139897427514785, −7.15576024923931373735185368729, −6.60311533599608811756224199124, −5.60603661368182320709479726436, −3.79239884410095802067850699850, −2.75531128747965607338796488580, −2.15222466683602588959249188618, 1.29298295527238399051943004688, 2.65700944160400447192490663477, 4.26097734204410416675861396163, 4.94758832672999984264911150930, 6.52255265429510025199285516384, 7.16113606393434282866593448667, 8.130782279648337196236388229706, 8.902989161135973426812215123721, 9.844237409722385703182190240172, 10.88647147366311296319805408955

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