↑

Properties

Label 2-507-13.12-c3-0-23
Degree $2$
Conductor $507$
Sign $-0.832 + 0.554i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.73i·2-s + 3·3-s − 5.95·4-s + 3.90i·5-s + 11.2i·6-s + 36.4i·7-s + 7.64i·8-s + 9·9-s − 14.5·10-s + 19.1i·11-s − 17.8·12-s − 136.·14-s + 11.7i·15-s − 76.1·16-s + 83.8·17-s + 33.6i·18-s + ⋯
L(s)  = 1  + 1.32i·2-s + 0.577·3-s − 0.744·4-s + 0.349i·5-s + 0.762i·6-s + 1.96i·7-s + 0.337i·8-s + 0.333·9-s − 0.461·10-s + 0.526i·11-s − 0.429·12-s − 2.59·14-s + 0.201i·15-s − 1.19·16-s + 1.19·17-s + 0.440i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.307478975\)
\(L(\frac12)\) \(\approx\) \(2.307478975\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 \)
good2 \( 1 - 3.73iT - 8T^{2} \)
5 \( 1 - 3.90iT - 125T^{2} \)
7 \( 1 - 36.4iT - 343T^{2} \)
11 \( 1 - 19.1iT - 1.33e3T^{2} \)
17 \( 1 - 83.8T + 4.91e3T^{2} \)
19 \( 1 + 46.8iT - 6.85e3T^{2} \)
23 \( 1 + 103.T + 1.21e4T^{2} \)
29 \( 1 - 108.T + 2.43e4T^{2} \)
31 \( 1 - 147. iT - 2.97e4T^{2} \)
37 \( 1 + 160. iT - 5.06e4T^{2} \)
41 \( 1 + 231. iT - 6.89e4T^{2} \)
43 \( 1 - 340.T + 7.95e4T^{2} \)
47 \( 1 - 119. iT - 1.03e5T^{2} \)
53 \( 1 + 732.T + 1.48e5T^{2} \)
59 \( 1 + 229. iT - 2.05e5T^{2} \)
61 \( 1 - 108.T + 2.26e5T^{2} \)
67 \( 1 + 10.3iT - 3.00e5T^{2} \)
71 \( 1 - 869. iT - 3.57e5T^{2} \)
73 \( 1 + 1.09e3iT - 3.89e5T^{2} \)
79 \( 1 - 140.T + 4.93e5T^{2} \)
83 \( 1 - 159. iT - 5.71e5T^{2} \)
89 \( 1 - 1.06e3iT - 7.04e5T^{2} \)
97 \( 1 + 858. iT - 9.12e5T^{2} \)
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

−11.00986626448184781025713386934, −9.706717754108462815701609116872, −8.943027543502981750069878132476, −8.271033774298136493848658311409, −7.43382573914592694533471068950, −6.43974074780646405702071836890, −5.64505379352995557098461649427, −4.79517769101509430253145422873, −3.03282209521565327086293403147, −2.09650779812442367028363088622, 0.68557339503161014955404970726, 1.45686915250863014864548454584, 3.04387100197728858728920436559, 3.81256625976026168368527786595, 4.60903844806288085072604051939, 6.35630845150016011434281594184, 7.49448556622402866625300714731, 8.204285085352290033186778473235, 9.524448181220348845779215162738, 10.13949183467537871829282745244

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