↑

Properties

Label 2-507-39.23-c0-0-0
Degree $2$
Conductor $507$
Sign $0.964 - 0.265i$
Analytic cond. $0.253025$
Root an. cond. $0.503016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.499 + 0.866i)9-s + 0.999·12-s + (−0.499 − 0.866i)16-s − 25-s − 0.999·27-s + (0.499 + 0.866i)36-s + (−1 + 1.73i)43-s + (0.499 − 0.866i)48-s + (−0.5 − 0.866i)49-s + (1 − 1.73i)61-s − 0.999·64-s + (−0.5 − 0.866i)75-s − 2·79-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.499 + 0.866i)9-s + 0.999·12-s + (−0.499 − 0.866i)16-s − 25-s − 0.999·27-s + (0.499 + 0.866i)36-s + (−1 + 1.73i)43-s + (0.499 − 0.866i)48-s + (−0.5 − 0.866i)49-s + (1 − 1.73i)61-s − 0.999·64-s + (−0.5 − 0.866i)75-s − 2·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.089558674\)
\(L(\frac12)\) \(\approx\) \(1.089558674\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−11.11886424873715407517357760874, −10.04713579963038427843560085613, −9.715119311473587752385605383509, −8.619170204653682386944463403602, −7.64778581848165550263809601842, −6.45163948241996375246769418140, −5.46928895015017093649496777937, −4.55170743068002638995567224754, −3.25510376772767607229686246654, −1.97836346568506537680213813821, 1.90836237988088961761687891822, 3.01664450915322485633931603272, 4.04570128725164222587001565108, 5.75358915284655606257774059667, 6.78226750023478199095015630063, 7.47486373456697764431656940157, 8.282063256949368370755606003582, 9.011768506709543287781317351895, 10.20845153979420460309348321019, 11.44895139159288995580079601776

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