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Properties

Label 2-250-125.106-c1-0-3
Degree $2$
Conductor $250$
Sign $-0.0672 - 0.997i$
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 + (−2.25 + 2.11i)3-s + (−0.425 − 0.904i)4-s + (2.21 + 0.332i)5-s + (0.578 + 3.03i)6-s + (−1.80 + 1.31i)7-s + (−0.992 − 0.125i)8-s + (0.410 − 6.52i)9-s + (1.46 − 1.68i)10-s + (−2.82 + 4.45i)11-s + (2.87 + 1.13i)12-s + (−0.371 + 5.90i)13-s + (0.140 + 2.22i)14-s + (−5.68 + 3.92i)15-s + (−0.637 + 0.770i)16-s + (−0.204 + 0.433i)17-s + ⋯
L(s)  = 1  + (0.378 − 0.597i)2-s + (−1.30 + 1.22i)3-s + (−0.212 − 0.452i)4-s + (0.988 + 0.148i)5-s + (0.236 + 1.23i)6-s + (−0.683 + 0.496i)7-s + (−0.350 − 0.0443i)8-s + (0.136 − 2.17i)9-s + (0.463 − 0.534i)10-s + (−0.853 + 1.34i)11-s + (0.829 + 0.328i)12-s + (−0.103 + 1.63i)13-s + (0.0374 + 0.595i)14-s + (−1.46 + 1.01i)15-s + (−0.159 + 0.192i)16-s + (−0.0494 + 0.105i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0672 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0672 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $-0.0672 - 0.997i$
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: \((2,\ 250,\ (\ :1/2),\ -0.0672 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583420 + 0.624082i\)
\(L(\frac12)\) \(\approx\) \(0.583420 + 0.624082i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.332i)T \)
good3 \( 1 + (2.25 - 2.11i)T + (0.188 - 2.99i)T^{2} \)
7 \( 1 + (1.80 - 1.31i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.82 - 4.45i)T + (-4.68 - 9.95i)T^{2} \)
13 \( 1 + (0.371 - 5.90i)T + (-12.8 - 1.62i)T^{2} \)
17 \( 1 + (0.204 - 0.433i)T + (-10.8 - 13.0i)T^{2} \)
19 \( 1 + (-0.678 - 0.636i)T + (1.19 + 18.9i)T^{2} \)
23 \( 1 + (1.45 + 0.373i)T + (20.1 + 11.0i)T^{2} \)
29 \( 1 + (-0.297 - 0.163i)T + (15.5 + 24.4i)T^{2} \)
31 \( 1 + (-4.71 + 10.0i)T + (-19.7 - 23.8i)T^{2} \)
37 \( 1 + (3.37 - 4.08i)T + (-6.93 - 36.3i)T^{2} \)
41 \( 1 + (0.698 - 0.179i)T + (35.9 - 19.7i)T^{2} \)
43 \( 1 + (-0.123 + 0.378i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-12.0 + 1.52i)T + (45.5 - 11.6i)T^{2} \)
53 \( 1 + (-1.22 + 6.41i)T + (-49.2 - 19.5i)T^{2} \)
59 \( 1 + (0.125 + 0.0498i)T + (43.0 + 40.3i)T^{2} \)
61 \( 1 + (-4.07 - 1.04i)T + (53.4 + 29.3i)T^{2} \)
67 \( 1 + (-4.70 + 2.58i)T + (35.9 - 56.5i)T^{2} \)
71 \( 1 + (-10.3 + 1.30i)T + (68.7 - 17.6i)T^{2} \)
73 \( 1 + (8.35 - 3.30i)T + (53.2 - 49.9i)T^{2} \)
79 \( 1 + (8.55 - 8.03i)T + (4.96 - 78.8i)T^{2} \)
83 \( 1 + (-1.90 - 1.79i)T + (5.21 + 82.8i)T^{2} \)
89 \( 1 + (-5.41 + 2.14i)T + (64.8 - 60.9i)T^{2} \)
97 \( 1 + (2.44 + 1.34i)T + (51.9 + 81.8i)T^{2} \)
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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

−12.11062391181556187482550235566, −11.36925081747818658508903460371, −10.24090903416507896847412230825, −9.841239146818833425074717326980, −9.177118717237646326645005319877, −6.77108853716346160273116915330, −5.93058468560859222031226465982, −5.00407194134685617064939404013, −4.11862672731835958235688905154, −2.31419909685075295440938616993, 0.69293168525854650503253703476, 2.91524967020559968163186994290, 5.24882845438158013710348122406, 5.71599675444114679837494236709, 6.55210537971950160585382613194, 7.50533896660422646660642687674, 8.539169569497073946989545786140, 10.28716141955861244543259309150, 10.80212828878326721183507268847, 12.20534225527926662347462557348

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