↑

Properties

Label 2-5e4-25.9-c1-0-30
Degree $2$
Conductor $625$
Sign $-0.425 + 0.904i$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
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  + (1.22 − 1.68i)2-s + (2.09 − 0.679i)3-s + (−0.725 − 2.23i)4-s + (1.41 − 4.36i)6-s − 0.992i·7-s + (−0.690 − 0.224i)8-s + (1.48 − 1.07i)9-s + (−1.61 − 1.17i)11-s + (−3.03 − 4.17i)12-s + (1.98 + 2.72i)13-s + (−1.67 − 1.21i)14-s + (2.57 − 1.87i)16-s + (−2.75 − 0.894i)17-s − 3.82i·18-s + (0.798 − 2.45i)19-s + ⋯
L(s)  = 1  + (0.866 − 1.19i)2-s + (1.20 − 0.392i)3-s + (−0.362 − 1.11i)4-s + (0.578 − 1.78i)6-s − 0.375i·7-s + (−0.244 − 0.0793i)8-s + (0.494 − 0.359i)9-s + (−0.487 − 0.354i)11-s + (−0.876 − 1.20i)12-s + (0.550 + 0.757i)13-s + (−0.447 − 0.325i)14-s + (0.643 − 0.467i)16-s + (−0.667 − 0.216i)17-s − 0.901i·18-s + (0.183 − 0.563i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.425 + 0.904i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -0.425 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77885 - 2.80302i\)
\(L(\frac12)\) \(\approx\) \(1.77885 - 2.80302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (-1.22 + 1.68i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.09 + 0.679i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + 0.992iT - 7T^{2} \)
11 \( 1 + (1.61 + 1.17i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.98 - 2.72i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.75 + 0.894i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.798 + 2.45i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.67 - 3.68i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.66 - 5.12i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.0421 + 0.129i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.26 + 1.73i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.98 - 5.07i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.64iT - 43T^{2} \)
47 \( 1 + (-9.44 + 3.06i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.19 - 2.33i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.97 + 2.89i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.24 - 1.62i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.07 + 0.675i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.97 - 9.17i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.456 + 0.627i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.89 + 15.0i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.68 + 0.547i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-11.7 - 8.52i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (16.1 - 5.26i)T + (78.4 - 57.0i)T^{2} \)
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

−10.59357636213359220109626438560, −9.521427482598419694786611648887, −8.707338739778624328921339063313, −7.78968738632187189339663719158, −6.84607909540070138374245111497, −5.41133312474197578079592454312, −4.28459882328894573276294730170, −3.39911681572691594578327376107, −2.55938411614520548804773964398, −1.51693799396447120878339073007, 2.33292268133123587224604046697, 3.54989422416281736010848682107, 4.36137820216969175601908239965, 5.47402900760099558391562199789, 6.28402324040061319220992501042, 7.40259082531596231456306119905, 8.234670632232654081084452764697, 8.682588055259508227665440831737, 9.910469378938301427034211255540, 10.63450057143759709048225000864

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