Properties

Label 2-50-25.6-c1-0-1
Degree $2$
Conductor $50$
Sign $0.890 - 0.455i$
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 + (1.09 − 0.792i)3-s + (−0.809 + 0.587i)4-s + (−2.15 − 0.587i)5-s + (1.09 + 0.792i)6-s − 0.833·7-s + (−0.809 − 0.587i)8-s + (−0.365 + 1.12i)9-s + (−0.107 − 2.23i)10-s + (0.257 + 0.792i)11-s + (−0.416 + 1.28i)12-s + (1.41 − 4.34i)13-s + (−0.257 − 0.792i)14-s + (−2.81 + 1.06i)15-s + (0.309 − 0.951i)16-s + (−4.41 − 3.20i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.629 − 0.457i)3-s + (−0.404 + 0.293i)4-s + (−0.964 − 0.262i)5-s + (0.445 + 0.323i)6-s − 0.314·7-s + (−0.286 − 0.207i)8-s + (−0.121 + 0.374i)9-s + (−0.0340 − 0.706i)10-s + (0.0776 + 0.238i)11-s + (−0.120 + 0.370i)12-s + (0.391 − 1.20i)13-s + (−0.0688 − 0.211i)14-s + (−0.727 + 0.275i)15-s + (0.0772 − 0.237i)16-s + (−1.06 − 0.777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :1/2),\ 0.890 - 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884801 + 0.213399i\)
\(L(\frac12)\) \(\approx\) \(0.884801 + 0.213399i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 + 0.587i)T \)
good3 \( 1 + (-1.09 + 0.792i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.833T + 7T^{2} \)
11 \( 1 + (-0.257 - 0.792i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.41 + 4.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.41 + 3.20i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-7.00 - 5.08i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.09 - 3.35i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.64 - 1.92i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.85 + 3.52i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.26 + 6.95i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.576 + 1.77i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.63T + 43T^{2} \)
47 \( 1 + (-0.674 + 0.489i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.19 - 3.77i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.18 + 12.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.81 - 5.59i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.21 + 0.881i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-1.91 + 1.38i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.02 + 3.16i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.18 - 3.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.97 - 7.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.16 + 6.66i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.97 - 6.51i)T + (29.9 - 92.2i)T^{2} \)
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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

−15.71722160923512237471653514967, −14.59849972747249108909689432482, −13.45810619216296466924212719357, −12.60639995224576316036787102613, −11.19477810397215719145609273934, −9.299343603771264096174189573705, −7.995955762103588709137229109224, −7.31881562226818636948363357107, −5.34535994999870021536395595042, −3.44266246576126949283524513030, 3.17459120572987956606991684069, 4.36420157987861449824327394935, 6.71674076957440159812442047772, 8.573252849835968570618815211044, 9.475582019359727795610839772923, 11.04574701177117349738944016356, 11.83181206204205240403486297782, 13.26576119433431699757004728734, 14.40079464679263799472507203816, 15.34091117881981262133671140841

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