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Properties

Label 2-250-25.16-c1-0-1
Degree $2$
Conductor $250$
Sign $-0.368 - 0.929i$
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  + (0.809 + 0.587i)2-s + (0.720 + 2.21i)3-s + (0.309 + 0.951i)4-s + (−0.720 + 2.21i)6-s − 3.77·7-s + (−0.309 + 0.951i)8-s + (−1.97 + 1.43i)9-s + (3.05 + 2.21i)11-s + (−1.88 + 1.37i)12-s + (2.56 − 1.86i)13-s + (−3.05 − 2.21i)14-s + (−0.809 + 0.587i)16-s + (0.430 − 1.32i)17-s − 2.44·18-s + (1.20 − 3.72i)19-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.416 + 1.28i)3-s + (0.154 + 0.475i)4-s + (−0.294 + 0.905i)6-s − 1.42·7-s + (−0.109 + 0.336i)8-s + (−0.658 + 0.478i)9-s + (0.920 + 0.668i)11-s + (−0.544 + 0.395i)12-s + (0.712 − 0.517i)13-s + (−0.816 − 0.592i)14-s + (−0.202 + 0.146i)16-s + (0.104 − 0.321i)17-s − 0.575·18-s + (0.277 − 0.853i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $-0.368 - 0.929i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{250} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 250,\ (\ :1/2),\ -0.368 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.983587 + 1.44808i\)
\(L(\frac12)\) \(\approx\) \(0.983587 + 1.44808i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 \)
good3 \( 1 + (-0.720 - 2.21i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 3.77T + 7T^{2} \)
11 \( 1 + (-3.05 - 2.21i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.56 + 1.86i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.430 + 1.32i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.20 + 3.72i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.720 - 0.523i)T + (7.10 + 21.8i)T^{2} \)
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 \( 1 + (-5.70 + 4.14i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.20 + 0.875i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.69T + 43T^{2} \)
47 \( 1 + (1.16 + 3.58i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.58 + 11.0i)T + (-42.8 + 31.1i)T^{2} \)
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 \( 1 + (4.73 - 14.5i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.06 + 6.36i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.18 - 3.03i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.558 + 1.71i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.08 + 9.47i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (11.7 + 8.52i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.0278 - 0.0857i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.57475110942515426250956437182, −11.38325301812141893759505611537, −10.23205341577466532102877734744, −9.447971562608465418359531080429, −8.773744163537623278110005242569, −7.18176644122828091594849701407, −6.24959109367676809035153294900, −4.95138023113316199912657723876, −3.82355646128898530067073586580, −3.08200669939636904610962311456, 1.33596109962438967145808493757, 2.92358458802301989313811056256, 3.95733881856060469958120193689, 6.13314788806585395608676479893, 6.39664081718093863555735580911, 7.66406687493828379542218596542, 8.893530205713066614593332269439, 9.800776828059633183270906468980, 11.10439032943599187050745496964, 12.10908340815484875007257544749

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