↑

Properties

Label 2-250-25.19-c1-0-0
Degree $2$
Conductor $250$
Sign $0.471 - 0.881i$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
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  + (−0.951 + 0.309i)2-s + (0.224 + 0.309i)3-s + (0.809 − 0.587i)4-s + (−0.309 − 0.224i)6-s + 3i·7-s + (−0.587 + 0.809i)8-s + (0.881 − 2.71i)9-s + (1.30 + 4.02i)11-s + (0.363 + 0.118i)12-s + (0.951 + 0.309i)13-s + (−0.927 − 2.85i)14-s + (0.309 − 0.951i)16-s + (−0.673 + 0.927i)17-s + 2.85i·18-s + (4.73 + 3.44i)19-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.129 + 0.178i)3-s + (0.404 − 0.293i)4-s + (−0.126 − 0.0916i)6-s + 1.13i·7-s + (−0.207 + 0.286i)8-s + (0.293 − 0.904i)9-s + (0.394 + 1.21i)11-s + (0.104 + 0.0340i)12-s + (0.263 + 0.0857i)13-s + (−0.247 − 0.762i)14-s + (0.0772 − 0.237i)16-s + (−0.163 + 0.224i)17-s + 0.672i·18-s + (1.08 + 0.789i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ 0.471 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.845515 + 0.506857i\)
\(L(\frac12)\) \(\approx\) \(0.845515 + 0.506857i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 \)
good3 \( 1 + (-0.224 - 0.309i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + (-1.30 - 4.02i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.951 - 0.309i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.673 - 0.927i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.73 - 3.44i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.67 + 0.545i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (7.66 - 5.56i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.190 - 0.138i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-7.91 - 2.57i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.454 + 1.40i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.23iT - 43T^{2} \)
47 \( 1 + (7.02 + 9.66i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (6.15 + 8.47i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.38 + 4.25i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.73 + 8.42i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-6.01 + 8.28i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-2.42 + 1.76i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (7.33 - 2.38i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.85 - 4.25i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.66 - 3.66i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.38 + 4.25i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.62 - 7.73i)T + (-29.9 + 92.2i)T^{2} \)
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

−12.13788621295179506774265643657, −11.30776618899777048077437159311, −9.888138874299703713599835208751, −9.408319223929727209621889614826, −8.527384376577953445343031061014, −7.30301270816615385471096574948, −6.33654639156622516688819371600, −5.16536624018022619918366350927, −3.53409756889326006433309838110, −1.81267306037006791034545598334, 1.09375105037554201008796845030, 2.96921416374152910795517659433, 4.35541275715992879365664468035, 5.96307092588452497747757970624, 7.30041475921198842461290573966, 7.85050013276357327995229524753, 9.054166169741118450814706527218, 9.969850823136729032062616988551, 11.16206603172656933828077475030, 11.30771095654010185812156847974

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