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Properties

Label 2-250-25.4-c1-0-3
Degree $2$
Conductor $250$
Sign $0.578 - 0.815i$
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  + (0.951 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ 0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57061 + 0.812085i\)
\(L(\frac12)\) \(\approx\) \(1.57061 + 0.812085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.62 - 7.73i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.04441364146372175895768574444, −11.54561215508141706491385522433, −10.48322797679371112222272716913, −9.206504352197674110791387541697, −8.266585293690629594823428012995, −7.12553653563684188436707300713, −5.77930953084154304377466049484, −5.21424529529663670703220538639, −3.71992476184533651231796234164, −2.31635545798386494666393888730, 1.45507448123385519112442549006, 3.47923890316461281412482621684, 4.38198616228745248864822649722, 5.70722251101934654261261355464, 7.05142796192003727642777851607, 7.45346278503030686299148037710, 9.363384305415434036358434587153, 10.05103399592924301024503130284, 11.08768430388181673892219677420, 12.22569747220744666928627161880

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