↑

Properties

Label 2-1503-1503.1168-c0-0-6
Degree $2$
Conductor $1503$
Sign $-0.0158 - 0.999i$
Analytic cond. $0.750094$
Root an. cond. $0.866080$
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  + (−0.723 + 1.25i)2-s + (0.580 + 0.814i)3-s + (−0.547 − 0.948i)4-s + (−1.44 + 0.137i)6-s + (0.786 − 1.36i)7-s + 0.137·8-s + (−0.327 + 0.945i)9-s + (0.888 − 1.53i)11-s + (0.454 − 0.996i)12-s + (1.13 + 1.97i)14-s + (0.447 − 0.775i)16-s + (−0.947 − 1.09i)18-s + 1.96·19-s + (1.56 − 0.149i)21-s + (1.28 + 2.22i)22-s + ⋯
L(s)  = 1  + (−0.723 + 1.25i)2-s + (0.580 + 0.814i)3-s + (−0.547 − 0.948i)4-s + (−1.44 + 0.137i)6-s + (0.786 − 1.36i)7-s + 0.137·8-s + (−0.327 + 0.945i)9-s + (0.888 − 1.53i)11-s + (0.454 − 0.996i)12-s + (1.13 + 1.97i)14-s + (0.447 − 0.775i)16-s + (−0.947 − 1.09i)18-s + 1.96·19-s + (1.56 − 0.149i)21-s + (1.28 + 2.22i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1503\)    =    \(3^{2} \cdot 167\)
Sign: $-0.0158 - 0.999i$
Analytic conductor: \(0.750094\)
Root analytic conductor: \(0.866080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1503} (1168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1503,\ (\ :0),\ -0.0158 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.015147615\)
\(L(\frac12)\) \(\approx\) \(1.015147615\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.580 - 0.814i)T \)
167 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.96T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.723 + 1.25i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.415 - 0.719i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.16T + T^{2} \)
97 \( 1 + (0.841 - 1.45i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.371486846191217346869930763103, −9.140812159280281927387314885656, −8.148661410400784270685805265279, −7.59621210892061704301455821347, −7.02312824297354387535002463420, −5.73434892477124580069238312127, −5.18832068463609215156959547568, −3.86318572216962843286778486148, −3.31522241300267039021032170370, −1.20605069411994647896840475068, 1.45925005899777842929935703909, 2.04941161487163597099571154830, 2.88593276033168685643316558949, 4.01481721201659192356451287771, 5.33244579141969058836416871709, 6.30673475589143743041089283588, 7.44030380738098892028008129389, 8.048529174721636007945014478230, 8.995460516485749079411412693569, 9.388979768580474166207208741950

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