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Properties

Label 2-50-25.21-c1-0-0
Degree $2$
Conductor $50$
Sign $0.637 - 0.770i$
Analytic cond. $0.399252$
Root an. cond. $0.631863$
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  + (−0.309 + 0.951i)2-s + (0.309 + 0.224i)3-s + (−0.809 − 0.587i)4-s + (1.80 + 1.31i)5-s + (−0.309 + 0.224i)6-s − 3·7-s + (0.809 − 0.587i)8-s + (−0.881 − 2.71i)9-s + (−1.80 + 1.31i)10-s + (1.30 − 4.02i)11-s + (−0.118 − 0.363i)12-s + (0.309 + 0.951i)13-s + (0.927 − 2.85i)14-s + (0.263 + 0.812i)15-s + (0.309 + 0.951i)16-s + (−0.927 + 0.673i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.178 + 0.129i)3-s + (−0.404 − 0.293i)4-s + (0.809 + 0.587i)5-s + (−0.126 + 0.0916i)6-s − 1.13·7-s + (0.286 − 0.207i)8-s + (−0.293 − 0.904i)9-s + (−0.572 + 0.415i)10-s + (0.394 − 1.21i)11-s + (−0.0340 − 0.104i)12-s + (0.0857 + 0.263i)13-s + (0.247 − 0.762i)14-s + (0.0681 + 0.209i)15-s + (0.0772 + 0.237i)16-s + (−0.224 + 0.163i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :1/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703195 + 0.330898i\)
\(L(\frac12)\) \(\approx\) \(0.703195 + 0.330898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-1.80 - 1.31i)T \)
good3 \( 1 + (-0.309 - 0.224i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + (-1.30 + 4.02i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.927 - 0.673i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.545 - 1.67i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-7.66 - 5.56i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.190 + 0.138i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.57 + 7.91i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.454 - 1.40i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 + (-9.66 - 7.02i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (8.47 + 6.15i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.38 + 4.25i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.73 - 8.42i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.28 + 6.01i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-2.42 - 1.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.38 + 7.33i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.85 - 4.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.66 + 2.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.38 + 4.25i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (7.73 + 5.62i)T + (29.9 + 92.2i)T^{2} \)
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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

−15.81662768346303265698832688894, −14.56511450325614585015439442035, −13.83724445695470717093770288508, −12.55047276469314785464892624560, −10.76405540278108253356939906848, −9.579639496543400602048680518587, −8.663096103605671259099197429464, −6.61173167876963624423268336093, −6.03090493458654367592850577096, −3.43402882809268056309322477990, 2.40102396990203361455319952814, 4.70857775135264936403522717060, 6.58734495133729244107940252311, 8.453699052979076533919704021011, 9.610167065681690100723965142551, 10.47219269246760589905403517313, 12.19592463867874050084188239272, 13.07879904460374119669649244903, 13.85836247374340955487040519745, 15.51422354601905256170928778297

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