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Properties

Label 2-250-25.21-c1-0-0
Degree $2$
Conductor $250$
Sign $-0.390 - 0.920i$
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 + (−2.34 − 1.70i)3-s + (−0.809 − 0.587i)4-s + (2.34 − 1.70i)6-s − 1.07·7-s + (0.809 − 0.587i)8-s + (1.67 + 5.15i)9-s + (−0.638 + 1.96i)11-s + (0.896 + 2.76i)12-s + (1.79 + 5.51i)13-s + (0.333 − 1.02i)14-s + (0.309 + 0.951i)16-s + (−0.666 + 0.484i)17-s − 5.42·18-s + (−2.76 + 2.00i)19-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−1.35 − 0.984i)3-s + (−0.404 − 0.293i)4-s + (0.958 − 0.696i)6-s − 0.407·7-s + (0.286 − 0.207i)8-s + (0.558 + 1.71i)9-s + (−0.192 + 0.592i)11-s + (0.258 + 0.796i)12-s + (0.496 + 1.52i)13-s + (0.0890 − 0.273i)14-s + (0.0772 + 0.237i)16-s + (−0.161 + 0.117i)17-s − 1.27·18-s + (−0.633 + 0.460i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $-0.390 - 0.920i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ -0.390 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.224531 + 0.339005i\)
\(L(\frac12)\) \(\approx\) \(0.224531 + 0.339005i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.34 + 1.70i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 1.07T + 7T^{2} \)
11 \( 1 + (0.638 - 1.96i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.79 - 5.51i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.666 - 0.484i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.76 - 2.00i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.606 - 1.86i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.847 - 0.616i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.71 - 1.24i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.68 - 8.27i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.03 + 6.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.47T + 43T^{2} \)
47 \( 1 + (6.99 + 5.08i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.07 + 1.50i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.45 - 4.47i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.86 - 8.82i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.52 + 2.56i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (3.96 + 2.88i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.401 + 1.23i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.82 - 4.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.1 - 8.79i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.61 - 11.1i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.413 + 0.300i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.33744790337746425041385406154, −11.52856953485541962608955446430, −10.57158536630862278052053428999, −9.452571090596547320148600273314, −8.193478447857407509764865450077, −6.98339774990152655160673233742, −6.54771092894529899975509468671, −5.56252447607295144011328948738, −4.33737840807449090638038776995, −1.66134306922778889926478198438, 0.41756539097360934869493111520, 3.13440360938131157882986120569, 4.36400332421360385316671949810, 5.49093703479943886321986074602, 6.34027547766429606129666247999, 8.037127595733344438182345734648, 9.259218957828849739589172850908, 10.16042606849787338962202058241, 10.88675022145357471010600668041, 11.34121239994048569268495067638

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