Properties

Label 2-250-25.21-c1-0-1
Degree $2$
Conductor $250$
Sign $-0.295 - 0.955i$
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 + (2.39 + 1.74i)3-s + (−0.809 − 0.587i)4-s + (−2.39 + 1.74i)6-s − 1.83·7-s + (0.809 − 0.587i)8-s + (1.79 + 5.51i)9-s + (−0.566 + 1.74i)11-s + (−0.916 − 2.82i)12-s + (0.747 + 2.29i)13-s + (0.566 − 1.74i)14-s + (0.309 + 0.951i)16-s + (2.25 − 1.63i)17-s − 5.79·18-s + (1.35 − 0.982i)19-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (1.38 + 1.00i)3-s + (−0.404 − 0.293i)4-s + (−0.979 + 0.711i)6-s − 0.692·7-s + (0.286 − 0.207i)8-s + (0.597 + 1.83i)9-s + (−0.170 + 0.525i)11-s + (−0.264 − 0.814i)12-s + (0.207 + 0.637i)13-s + (0.151 − 0.466i)14-s + (0.0772 + 0.237i)16-s + (0.546 − 0.396i)17-s − 1.36·18-s + (0.310 − 0.225i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $-0.295 - 0.955i$
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: \((2,\ 250,\ (\ :1/2),\ -0.295 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.920166 + 1.24716i\)
\(L(\frac12)\) \(\approx\) \(0.920166 + 1.24716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.39 - 1.74i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 1.83T + 7T^{2} \)
11 \( 1 + (0.566 - 1.74i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.747 - 2.29i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.25 + 1.63i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.35 + 0.982i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.39 + 7.38i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (6.13 + 4.45i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.28 + 3.11i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.406 - 1.25i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.08 - 3.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 + (-1.48 - 1.07i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.27 + 3.83i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.79 + 8.61i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.799 + 2.46i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-7.68 + 5.58i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.247 + 0.179i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.61 - 14.2i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.79 - 2.03i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.15 - 3.74i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.02 - 3.15i)T + (-72.0 - 52.3i)T^{2} \)
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

−12.66337760042203289078405407156, −11.06165289010058114957162349562, −9.764160682833010905819186751081, −9.590921199057144384847616078652, −8.527312234921608128648332566595, −7.66203998050267811863729521205, −6.48474609328838122576338925580, −4.90232385750024466615905558533, −3.90845368227141001799821572823, −2.60215018890690687716674760333, 1.38716800145418134770071205111, 2.94956504348553795740003244534, 3.57061511038160861522988865433, 5.74455353124308980761452452090, 7.19943047783020608516036587457, 7.947501618918807556273871096731, 8.897793984069911952467757064752, 9.622173993053082042858028285918, 10.76608781990733517830027007764, 12.04760809341711242206727581291

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