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Properties

Label 2-250-25.6-c1-0-1
Degree $2$
Conductor $250$
Sign $0.738 - 0.673i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−1.09 + 0.792i)3-s + (−0.809 + 0.587i)4-s + (1.09 + 0.792i)6-s + 0.833·7-s + (0.809 + 0.587i)8-s + (−0.365 + 1.12i)9-s + (0.257 + 0.792i)11-s + (0.416 − 1.28i)12-s + (−1.41 + 4.34i)13-s + (−0.257 − 0.792i)14-s + (0.309 − 0.951i)16-s + (4.41 + 3.20i)17-s + 1.18·18-s + (7.00 + 5.08i)19-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.629 + 0.457i)3-s + (−0.404 + 0.293i)4-s + (0.445 + 0.323i)6-s + 0.314·7-s + (0.286 + 0.207i)8-s + (−0.121 + 0.374i)9-s + (0.0776 + 0.238i)11-s + (0.120 − 0.370i)12-s + (−0.391 + 1.20i)13-s + (−0.0688 − 0.211i)14-s + (0.0772 − 0.237i)16-s + (1.06 + 0.777i)17-s + 0.278·18-s + (1.60 + 1.16i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.738 - 0.673i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ 0.738 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.775178 + 0.300359i\)
\(L(\frac12)\) \(\approx\) \(0.775178 + 0.300359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 \)
good3 \( 1 + (1.09 - 0.792i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 0.833T + 7T^{2} \)
11 \( 1 + (-0.257 - 0.792i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.41 - 4.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.41 - 3.20i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-7.00 - 5.08i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.09 + 3.35i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.64 - 1.92i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.85 + 3.52i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.26 - 6.95i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.576 + 1.77i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.63T + 43T^{2} \)
47 \( 1 + (0.674 - 0.489i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-5.19 + 3.77i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.18 + 12.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.81 - 5.59i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.21 - 0.881i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-1.91 + 1.38i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.02 - 3.16i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.18 - 3.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.97 + 7.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.16 + 6.66i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-8.97 + 6.51i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.84214878724717565089456484806, −11.35171640956104479310116815148, −10.20311143231574897433036255053, −9.704933802015679033119885183027, −8.377752922582894151150850287253, −7.38811298797780700116078359080, −5.80752209975505583333224687717, −4.81995417331240497044736845515, −3.63714236362499115810375396189, −1.79789368096847621604897229514, 0.829476121908019914870984933973, 3.28388600180850106421913167291, 5.23791464244498667957547788080, 5.69738965596457730675705957871, 7.16056165634872631683535733936, 7.64396976299781020773896353016, 9.031134561735626369758096981428, 9.870836623255152380690895909037, 11.13267072375448028448553524742, 11.91510678267978330261035762810

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