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Properties

Label 2-250-25.14-c1-0-5
Degree $2$
Conductor $250$
Sign $0.596 + 0.802i$
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 + (−2.21 − 0.720i)3-s + (−0.309 + 0.951i)4-s + (−0.720 − 2.21i)6-s − 3.77i·7-s + (−0.951 + 0.309i)8-s + (1.97 + 1.43i)9-s + (3.05 − 2.21i)11-s + (1.37 − 1.88i)12-s + (1.86 − 2.56i)13-s + (3.05 − 2.21i)14-s + (−0.809 − 0.587i)16-s + (−1.32 + 0.430i)17-s + 2.44i·18-s + (−1.20 − 3.72i)19-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−1.28 − 0.416i)3-s + (−0.154 + 0.475i)4-s + (−0.294 − 0.905i)6-s − 1.42i·7-s + (−0.336 + 0.109i)8-s + (0.658 + 0.478i)9-s + (0.920 − 0.668i)11-s + (0.395 − 0.544i)12-s + (0.517 − 0.712i)13-s + (0.816 − 0.592i)14-s + (−0.202 − 0.146i)16-s + (−0.321 + 0.104i)17-s + 0.575i·18-s + (−0.277 − 0.853i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $0.596 + 0.802i$
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: \((2,\ 250,\ (\ :1/2),\ 0.596 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.822022 - 0.412951i\)
\(L(\frac12)\) \(\approx\) \(0.822022 - 0.412951i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.720i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + 3.77iT - 7T^{2} \)
11 \( 1 + (-3.05 + 2.21i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.86 + 2.56i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.32 - 0.430i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.20 + 3.72i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.523 + 0.720i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.0152 - 0.0468i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.72 + 5.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.14 - 5.70i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.20 - 0.875i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.69iT - 43T^{2} \)
47 \( 1 + (3.58 + 1.16i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-11.0 - 3.58i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.558 + 0.405i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (8.38 - 6.08i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-14.5 + 4.73i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.06 - 6.36i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.03 + 4.18i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.558 + 1.71i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-9.47 + 3.08i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-11.7 + 8.52i)T + (27.5 - 84.6i)T^{2} \)
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

−11.84277845096250510378598987423, −11.13900898964678408236321240791, −10.35683284032033133609978354543, −8.832155023995156143900993488119, −7.57886070971946121539528218671, −6.65804900539026882670092901637, −6.03133854745525774286942923227, −4.77769099386334999692154461733, −3.64264507201996091049429410286, −0.78862573505396361659707561012, 1.94719516848158823379694469843, 3.86503049987923252367988137652, 5.01855222291866623387666587833, 5.87396775091056875070822458726, 6.71439309274277347858601587555, 8.684638288838712472065741232807, 9.504130658365756565533760991667, 10.57416718055177393013886305896, 11.46383330955328933924967527252, 12.07270044182832398419360951343

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