↑

Properties

Label 2-975-39.17-c0-0-0
Degree $2$
Conductor $975$
Sign $-0.872 - 0.488i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
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 + (−1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s + 13-s + (−0.499 + 0.866i)16-s + (−1.5 + 0.866i)19-s − 1.73i·21-s + 0.999·27-s + (−1.5 − 0.866i)28-s + (0.499 − 0.866i)36-s + (−0.5 + 0.866i)39-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)48-s + (1 − 1.73i)49-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 0.999·12-s + 13-s + (−0.499 + 0.866i)16-s + (−1.5 + 0.866i)19-s − 1.73i·21-s + 0.999·27-s + (−1.5 − 0.866i)28-s + (0.499 − 0.866i)36-s + (−0.5 + 0.866i)39-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)48-s + (1 − 1.73i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.872 - 0.488i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.872 - 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6832076425\)
\(L(\frac12)\) \(\approx\) \(0.6832076425\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 - T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - T + T^{2} \)
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

−10.51765468841867645645644145286, −9.818694976065765683003596648741, −8.791037834137066813027696600432, −8.426514500827832571216009803274, −6.87816176147505392195646336754, −6.27159936783007483939384332111, −5.60152252350275321508479302945, −4.04866188662035645577902502127, −3.50484480480085403486912670265, −2.44510462293072115045458923242, 0.66061533767724240031623338626, 2.08624037924958537313003745850, 3.35510042825608865533808537572, 4.71699370965639685397361418056, 5.94612061629038566264218035499, 6.55207435947450918821557432695, 6.87061299290143230781790529117, 8.032530829458682620797121120378, 9.152383049186592932706878777873, 10.04375524246975758218060622369

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