Properties

Label 2-50-5.4-c21-0-26
Degree $2$
Conductor $50$
Sign $0.894 + 0.447i$
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 + 2.19e4i·3-s − 1.04e6·4-s − 2.24e7·6-s − 7.22e8i·7-s − 1.07e9i·8-s + 9.97e9·9-s + 4.99e10·11-s − 2.29e10i·12-s + 2.43e10i·13-s + 7.40e11·14-s + 1.09e12·16-s − 5.76e12i·17-s + 1.02e13i·18-s + 3.02e13·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.214i·3-s − 0.5·4-s − 0.151·6-s − 0.967i·7-s − 0.353i·8-s + 0.954·9-s + 0.580·11-s − 0.107i·12-s + 0.0489i·13-s + 0.683·14-s + 0.250·16-s − 0.694i·17-s + 0.674i·18-s + 1.13·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
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: \((2,\ 50,\ (\ :21/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.142203907\)
\(L(\frac12)\) \(\approx\) \(2.142203907\)
\(L(\frac{23}{2})\) 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 \( 1 - 2.19e4iT - 1.04e10T^{2} \)
7 \( 1 + 7.22e8iT - 5.58e17T^{2} \)
11 \( 1 - 4.99e10T + 7.40e21T^{2} \)
13 \( 1 - 2.43e10iT - 2.47e23T^{2} \)
17 \( 1 + 5.76e12iT - 6.90e25T^{2} \)
19 \( 1 - 3.02e13T + 7.14e26T^{2} \)
23 \( 1 - 1.45e14iT - 3.94e28T^{2} \)
29 \( 1 - 1.16e15T + 5.13e30T^{2} \)
31 \( 1 + 7.43e15T + 2.08e31T^{2} \)
37 \( 1 + 5.46e16iT - 8.55e32T^{2} \)
41 \( 1 + 2.08e16T + 7.38e33T^{2} \)
43 \( 1 + 7.61e16iT - 2.00e34T^{2} \)
47 \( 1 - 5.08e17iT - 1.30e35T^{2} \)
53 \( 1 + 1.34e18iT - 1.62e36T^{2} \)
59 \( 1 + 2.30e18T + 1.54e37T^{2} \)
61 \( 1 - 5.26e18T + 3.10e37T^{2} \)
67 \( 1 + 1.09e19iT - 2.22e38T^{2} \)
71 \( 1 + 1.33e19T + 7.52e38T^{2} \)
73 \( 1 - 1.79e19iT - 1.34e39T^{2} \)
79 \( 1 + 1.18e20T + 7.08e39T^{2} \)
83 \( 1 + 1.34e20iT - 1.99e40T^{2} \)
89 \( 1 + 3.47e20T + 8.65e40T^{2} \)
97 \( 1 + 3.76e20iT - 5.27e41T^{2} \)
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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

−11.19477130178134744118842036793, −9.953544504111557726251248356989, −9.142283332375504660123348635321, −7.48989315666565868412558606718, −7.04425102208540116385835235577, −5.53126476269469219251245594784, −4.35489133392152480753343372744, −3.48679391756447898342699087378, −1.52492512125284861820082996786, −0.48711943026516728961471297607, 1.05059574562363174625931959825, 1.90405425739819218909386414842, 3.12953887969552196056062404792, 4.33665920454942775348907868998, 5.58210506596827485326192348947, 6.89412319164144661292016334381, 8.306753121544125369515706806953, 9.357687428841857827189815113083, 10.34466416286412881529845455417, 11.67619330162558354084496742575

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