Properties

Label 2-50-5.4-c21-0-15
Degree $2$
Conductor $50$
Sign $0.447 + 0.894i$
Analytic cond. $139.738$
Root an. cond. $11.8211$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.02e3i·2-s + 1.85e5i·3-s − 1.04e6·4-s + 1.90e8·6-s − 4.88e8i·7-s + 1.07e9i·8-s − 2.40e10·9-s − 7.28e10·11-s − 1.94e11i·12-s + 9.02e11i·13-s − 4.99e11·14-s + 1.09e12·16-s + 6.32e12i·17-s + 2.46e13i·18-s − 4.36e13·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.81i·3-s − 0.5·4-s + 1.28·6-s − 0.653i·7-s + 0.353i·8-s − 2.30·9-s − 0.846·11-s − 0.908i·12-s + 1.81i·13-s − 0.461·14-s + 0.250·16-s + 0.760i·17-s + 1.62i·18-s − 1.63·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(139.738\)
Root analytic conductor: \(11.8211\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :21/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.1756855648\)
\(L(\frac12)\) \(\approx\) \(0.1756855648\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.02e3iT \)
5 \( 1 \)
good3 \( 1 - 1.85e5iT - 1.04e10T^{2} \)
7 \( 1 + 4.88e8iT - 5.58e17T^{2} \)
11 \( 1 + 7.28e10T + 7.40e21T^{2} \)
13 \( 1 - 9.02e11iT - 2.47e23T^{2} \)
17 \( 1 - 6.32e12iT - 6.90e25T^{2} \)
19 \( 1 + 4.36e13T + 7.14e26T^{2} \)
23 \( 1 + 9.00e13iT - 3.94e28T^{2} \)
29 \( 1 - 1.34e15T + 5.13e30T^{2} \)
31 \( 1 + 4.56e14T + 2.08e31T^{2} \)
37 \( 1 - 2.84e16iT - 8.55e32T^{2} \)
41 \( 1 + 1.42e17T + 7.38e33T^{2} \)
43 \( 1 - 1.47e17iT - 2.00e34T^{2} \)
47 \( 1 + 1.71e17iT - 1.30e35T^{2} \)
53 \( 1 - 1.36e18iT - 1.62e36T^{2} \)
59 \( 1 + 6.60e18T + 1.54e37T^{2} \)
61 \( 1 - 1.48e18T + 3.10e37T^{2} \)
67 \( 1 + 2.80e19iT - 2.22e38T^{2} \)
71 \( 1 + 4.31e19T + 7.52e38T^{2} \)
73 \( 1 - 2.89e17iT - 1.34e39T^{2} \)
79 \( 1 - 5.22e19T + 7.08e39T^{2} \)
83 \( 1 + 5.91e19iT - 1.99e40T^{2} \)
89 \( 1 - 4.79e20T + 8.65e40T^{2} \)
97 \( 1 + 8.16e20iT - 5.27e41T^{2} \)
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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

−10.74141771660538279439203848255, −10.37887175083548242891666404250, −9.243682524835124540251662545937, −8.353796671504688903716930249267, −6.31909421526558432980876246254, −4.68350718035200295563443065006, −4.31113400165147675702993184534, −3.21199514285462653976247994484, −1.91599373742403515771393925536, −0.05081584552515379612103701767, 0.65013746493795918596145825289, 2.07980996568353320145676973922, 3.00988520159584953203531880765, 5.25268897648368828027180591454, 5.99073796817444288330856780288, 7.11035132011884550545396479556, 7.993736184524541857816320411120, 8.687759748702197804934877019419, 10.54105184001769903769579446174, 12.05671047064175990649041102492

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