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Properties

Label 2-250-25.14-c1-0-3
Degree $2$
Conductor $250$
Sign $0.236 - 0.971i$
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  + (0.587 + 0.809i)2-s + (1.63 + 0.529i)3-s + (−0.309 + 0.951i)4-s + (0.529 + 1.63i)6-s + 2.77i·7-s + (−0.951 + 0.309i)8-s + (−0.0483 − 0.0351i)9-s + (−2.24 + 1.63i)11-s + (−1.00 + 1.38i)12-s + (3.33 − 4.59i)13-s + (−2.24 + 1.63i)14-s + (−0.809 − 0.587i)16-s + (4.90 − 1.59i)17-s − 0.0597i·18-s + (−0.436 − 1.34i)19-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (0.941 + 0.305i)3-s + (−0.154 + 0.475i)4-s + (0.216 + 0.665i)6-s + 1.04i·7-s + (−0.336 + 0.109i)8-s + (−0.0161 − 0.0117i)9-s + (−0.676 + 0.491i)11-s + (−0.290 + 0.400i)12-s + (0.925 − 1.27i)13-s + (−0.599 + 0.435i)14-s + (−0.202 − 0.146i)16-s + (1.18 − 0.386i)17-s − 0.0140i·18-s + (−0.100 − 0.308i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.236 - 0.971i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ 0.236 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50832 + 1.18504i\)
\(L(\frac12)\) \(\approx\) \(1.50832 + 1.18504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 \)
good3 \( 1 + (-1.63 - 0.529i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 - 2.77iT - 7T^{2} \)
11 \( 1 + (2.24 - 1.63i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.33 + 4.59i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.90 + 1.59i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.436 + 1.34i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.384 - 0.529i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.26 - 3.89i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.20 + 6.77i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.615 + 0.847i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.36 + 5.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.24iT - 43T^{2} \)
47 \( 1 + (-2.63 - 0.857i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.500 + 0.162i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.05 + 2.22i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.76 + 6.36i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.11 - 1.33i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (4.09 - 12.5i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.47 + 3.40i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.05 + 9.41i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.44 - 1.44i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.43 - 5.39i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.0857 - 0.0278i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.55669676545750465333350057315, −11.45954399508187894351994227594, −10.11755670514339289804612303999, −9.126893105789744266293587173959, −8.304616456388957276923137547699, −7.56721153066673030319523139183, −5.96140979742995332399300065316, −5.21050172565988576172203699795, −3.56806534921689257671207495372, −2.65598387754391790833711282039, 1.60020845950139963637205022502, 3.17026364367708987197616428163, 4.06395489624930904649594546227, 5.59289821902666988540158210165, 6.95626166111199706498587476683, 8.061565924566752054720789960917, 8.887637225914659669739489552660, 10.12527159935841309249572018512, 10.88262510157072864964111049143, 11.88123281650408947871914264002

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