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Properties

Label 2-250-125.106-c1-0-5
Degree $2$
Conductor $250$
Sign $0.679 - 0.733i$
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  + (−0.535 + 0.844i)2-s + (−1.66 + 1.56i)3-s + (−0.425 − 0.904i)4-s + (1.81 − 1.31i)5-s + (−0.427 − 2.24i)6-s + (3.35 − 2.43i)7-s + (0.992 + 0.125i)8-s + (0.138 − 2.19i)9-s + (0.136 + 2.23i)10-s + (2.18 − 3.44i)11-s + (2.12 + 0.839i)12-s + (−0.362 + 5.75i)13-s + (0.260 + 4.14i)14-s + (−0.964 + 5.00i)15-s + (−0.637 + 0.770i)16-s + (−0.879 + 1.86i)17-s + ⋯
L(s)  = 1  + (−0.378 + 0.597i)2-s + (−0.959 + 0.901i)3-s + (−0.212 − 0.452i)4-s + (0.810 − 0.586i)5-s + (−0.174 − 0.914i)6-s + (1.26 − 0.921i)7-s + (0.350 + 0.0443i)8-s + (0.0460 − 0.732i)9-s + (0.0430 + 0.705i)10-s + (0.658 − 1.03i)11-s + (0.612 + 0.242i)12-s + (−0.100 + 1.59i)13-s + (0.0696 + 1.10i)14-s + (−0.249 + 1.29i)15-s + (−0.159 + 0.192i)16-s + (−0.213 + 0.453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.679 - 0.733i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ 0.679 - 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898689 + 0.392612i\)
\(L(\frac12)\) \(\approx\) \(0.898689 + 0.392612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.535 - 0.844i)T \)
5 \( 1 + (-1.81 + 1.31i)T \)
good3 \( 1 + (1.66 - 1.56i)T + (0.188 - 2.99i)T^{2} \)
7 \( 1 + (-3.35 + 2.43i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-2.18 + 3.44i)T + (-4.68 - 9.95i)T^{2} \)
13 \( 1 + (0.362 - 5.75i)T + (-12.8 - 1.62i)T^{2} \)
17 \( 1 + (0.879 - 1.86i)T + (-10.8 - 13.0i)T^{2} \)
19 \( 1 + (3.95 + 3.71i)T + (1.19 + 18.9i)T^{2} \)
23 \( 1 + (-7.48 - 1.92i)T + (20.1 + 11.0i)T^{2} \)
29 \( 1 + (-6.86 - 3.77i)T + (15.5 + 24.4i)T^{2} \)
31 \( 1 + (0.436 - 0.928i)T + (-19.7 - 23.8i)T^{2} \)
37 \( 1 + (0.297 - 0.359i)T + (-6.93 - 36.3i)T^{2} \)
41 \( 1 + (2.75 - 0.708i)T + (35.9 - 19.7i)T^{2} \)
43 \( 1 + (0.243 - 0.749i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-5.16 + 0.652i)T + (45.5 - 11.6i)T^{2} \)
53 \( 1 + (0.582 - 3.05i)T + (-49.2 - 19.5i)T^{2} \)
59 \( 1 + (-0.853 - 0.337i)T + (43.0 + 40.3i)T^{2} \)
61 \( 1 + (9.82 + 2.52i)T + (53.4 + 29.3i)T^{2} \)
67 \( 1 + (10.9 - 6.03i)T + (35.9 - 56.5i)T^{2} \)
71 \( 1 + (3.88 - 0.490i)T + (68.7 - 17.6i)T^{2} \)
73 \( 1 + (6.24 - 2.47i)T + (53.2 - 49.9i)T^{2} \)
79 \( 1 + (-4.11 + 3.86i)T + (4.96 - 78.8i)T^{2} \)
83 \( 1 + (10.6 + 9.97i)T + (5.21 + 82.8i)T^{2} \)
89 \( 1 + (-14.7 + 5.82i)T + (64.8 - 60.9i)T^{2} \)
97 \( 1 + (10.7 + 5.91i)T + (51.9 + 81.8i)T^{2} \)
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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

−11.77097739109319682574023164768, −10.97738207918766920368492298510, −10.44470124696732241795964458070, −9.124617096263533221877933463478, −8.629474328253957337781391089883, −6.99404239134304078343130989138, −6.06653062723193617677159511222, −4.83374121992359875235965015366, −4.41620695706320722708379196392, −1.32502040882720008578664227554, 1.44068694012457465648293644432, 2.58818583474549004460818450182, 4.87338056801092939968881510713, 5.85451373190819038455087409322, 6.92056396879297410883894122647, 7.978573887107607518972126376598, 9.109189296365693909217732040015, 10.34693563372270767832572509505, 11.00817263263721798362253601637, 12.01996969362007243777743638086

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