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Properties

Label 2-250-125.106-c1-0-2
Degree $2$
Conductor $250$
Sign $-0.813 - 0.581i$
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 + (0.174 − 0.163i)3-s + (−0.425 − 0.904i)4-s + (−2.21 + 0.323i)5-s + (0.0447 + 0.234i)6-s + (0.0517 − 0.0376i)7-s + (0.992 + 0.125i)8-s + (−0.184 + 2.93i)9-s + (0.912 − 2.04i)10-s + (−2.94 + 4.63i)11-s + (−0.221 − 0.0878i)12-s + (−0.141 + 2.24i)13-s + (0.00401 + 0.0638i)14-s + (−0.332 + 0.417i)15-s + (−0.637 + 0.770i)16-s + (−1.23 + 2.61i)17-s + ⋯
L(s)  = 1  + (−0.378 + 0.597i)2-s + (0.100 − 0.0943i)3-s + (−0.212 − 0.452i)4-s + (−0.989 + 0.144i)5-s + (0.0182 + 0.0957i)6-s + (0.0195 − 0.0142i)7-s + (0.350 + 0.0443i)8-s + (−0.0615 + 0.979i)9-s + (0.288 − 0.645i)10-s + (−0.887 + 1.39i)11-s + (−0.0640 − 0.0253i)12-s + (−0.0392 + 0.623i)13-s + (0.00107 + 0.0170i)14-s + (−0.0857 + 0.107i)15-s + (−0.159 + 0.192i)16-s + (−0.298 + 0.635i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250\)    =    \(2 \cdot 5^{3}\)
Sign: $-0.813 - 0.581i$
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: \((2,\ 250,\ (\ :1/2),\ -0.813 - 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.179813 + 0.561161i\)
\(L(\frac12)\) \(\approx\) \(0.179813 + 0.561161i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.323i)T \)
good3 \( 1 + (-0.174 + 0.163i)T + (0.188 - 2.99i)T^{2} \)
7 \( 1 + (-0.0517 + 0.0376i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.94 - 4.63i)T + (-4.68 - 9.95i)T^{2} \)
13 \( 1 + (0.141 - 2.24i)T + (-12.8 - 1.62i)T^{2} \)
17 \( 1 + (1.23 - 2.61i)T + (-10.8 - 13.0i)T^{2} \)
19 \( 1 + (5.39 + 5.06i)T + (1.19 + 18.9i)T^{2} \)
23 \( 1 + (-8.71 - 2.23i)T + (20.1 + 11.0i)T^{2} \)
29 \( 1 + (6.52 + 3.58i)T + (15.5 + 24.4i)T^{2} \)
31 \( 1 + (1.99 - 4.23i)T + (-19.7 - 23.8i)T^{2} \)
37 \( 1 + (-2.63 + 3.18i)T + (-6.93 - 36.3i)T^{2} \)
41 \( 1 + (-0.262 + 0.0672i)T + (35.9 - 19.7i)T^{2} \)
43 \( 1 + (-1.94 + 5.98i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.85 - 0.234i)T + (45.5 - 11.6i)T^{2} \)
53 \( 1 + (1.73 - 9.08i)T + (-49.2 - 19.5i)T^{2} \)
59 \( 1 + (-3.57 - 1.41i)T + (43.0 + 40.3i)T^{2} \)
61 \( 1 + (-11.3 - 2.92i)T + (53.4 + 29.3i)T^{2} \)
67 \( 1 + (-13.4 + 7.36i)T + (35.9 - 56.5i)T^{2} \)
71 \( 1 + (-11.3 + 1.43i)T + (68.7 - 17.6i)T^{2} \)
73 \( 1 + (7.70 - 3.05i)T + (53.2 - 49.9i)T^{2} \)
79 \( 1 + (-2.59 + 2.44i)T + (4.96 - 78.8i)T^{2} \)
83 \( 1 + (2.32 + 2.18i)T + (5.21 + 82.8i)T^{2} \)
89 \( 1 + (7.76 - 3.07i)T + (64.8 - 60.9i)T^{2} \)
97 \( 1 + (14.3 + 7.89i)T + (51.9 + 81.8i)T^{2} \)
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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

−12.66496564245697884603777428094, −11.12662812451124181371184626630, −10.72329186785135677693654319144, −9.349678550052039578382290224951, −8.393780427733465167524125621524, −7.41947549105074093060709621548, −6.91021866793400624217524061708, −5.14724925891845809873933105983, −4.30311518151964147322794215783, −2.29509023120132211493153607521, 0.51407956240021504156579888916, 2.96786211092966970771506492165, 3.84720356155326815032610310410, 5.34427743463925692112775021702, 6.83314063894527748952647811658, 8.150050370027159906936574880327, 8.615550860166169952519384503679, 9.781787358665167409419332682179, 11.06895093175838122248831898652, 11.30316963611683002050614776554

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