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Properties

Label 2-250-25.4-c1-0-1
Degree $2$
Conductor $250$
Sign $-0.753 - 0.657i$
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 + (−1.74 + 2.39i)3-s + (0.809 + 0.587i)4-s + (−2.39 + 1.74i)6-s + 1.83i·7-s + (0.587 + 0.809i)8-s + (−1.79 − 5.51i)9-s + (−0.566 + 1.74i)11-s + (−2.82 + 0.916i)12-s + (−2.29 + 0.747i)13-s + (−0.566 + 1.74i)14-s + (0.309 + 0.951i)16-s + (−1.63 − 2.25i)17-s − 5.79i·18-s + (−1.35 + 0.982i)19-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (−1.00 + 1.38i)3-s + (0.404 + 0.293i)4-s + (−0.979 + 0.711i)6-s + 0.692i·7-s + (0.207 + 0.286i)8-s + (−0.597 − 1.83i)9-s + (−0.170 + 0.525i)11-s + (−0.814 + 0.264i)12-s + (−0.637 + 0.207i)13-s + (−0.151 + 0.466i)14-s + (0.0772 + 0.237i)16-s + (−0.396 − 0.546i)17-s − 1.36i·18-s + (−0.310 + 0.225i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $-0.753 - 0.657i$
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: \((2,\ 250,\ (\ :1/2),\ -0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421938 + 1.12485i\)
\(L(\frac12)\) \(\approx\) \(0.421938 + 1.12485i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.74 - 2.39i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 - 1.83iT - 7T^{2} \)
11 \( 1 + (0.566 - 1.74i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.29 - 0.747i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.63 + 2.25i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.35 - 0.982i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-7.38 - 2.39i)T + (18.6 + 13.5i)T^{2} \)
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 \( 1 + (-1.25 + 0.406i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.08 - 3.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.30iT - 43T^{2} \)
47 \( 1 + (-1.07 + 1.48i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.83 + 5.27i)T + (-16.3 - 50.4i)T^{2} \)
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 \( 1 + (5.58 + 7.68i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (0.247 + 0.179i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (14.2 + 4.61i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.79 + 2.03i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.74 + 5.15i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.02 + 3.15i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.51 + 8.97i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.15197827845445438190664889433, −11.57779261377142157408309059848, −10.63865785586426520494164633362, −9.730125014651645041054779800635, −8.802466775895459408236116919477, −7.13096899655543069337695349716, −6.01840232293526279753357237853, −5.02368471137612453205341992813, −4.46262947078640864084646765576, −2.91124705947363751108686027866, 0.903479216928537923028048920257, 2.60955446306726876295977096153, 4.51485445617811514168376029058, 5.63839864058397712403390606016, 6.63598176751929927590292936106, 7.27680334421857995041512533952, 8.475090598953481752877119696496, 10.33348551291694746146518290939, 10.98665515168514857532566113168, 11.88538563298662469417262060962

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