Properties

Label 2-50-5.4-c21-0-24
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 + 2.64e4i·3-s − 1.04e6·4-s + 2.70e7·6-s − 3.30e8i·7-s + 1.07e9i·8-s + 9.76e9·9-s + 1.32e9·11-s − 2.77e10i·12-s − 1.22e11i·13-s − 3.38e11·14-s + 1.09e12·16-s − 1.18e12i·17-s − 9.99e12i·18-s − 1.96e13·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.258i·3-s − 0.5·4-s + 0.182·6-s − 0.442i·7-s + 0.353i·8-s + 0.933·9-s + 0.0154·11-s − 0.129i·12-s − 0.246i·13-s − 0.313·14-s + 0.250·16-s − 0.142i·17-s − 0.659i·18-s − 0.734·19-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

\(L(11)\) \(\approx\) \(1.894400465\)
\(L(\frac12)\) \(\approx\) \(1.894400465\)
\(L(\frac{23}{2})\) 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 \( 1 - 2.64e4iT - 1.04e10T^{2} \)
7 \( 1 + 3.30e8iT - 5.58e17T^{2} \)
11 \( 1 - 1.32e9T + 7.40e21T^{2} \)
13 \( 1 + 1.22e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.18e12iT - 6.90e25T^{2} \)
19 \( 1 + 1.96e13T + 7.14e26T^{2} \)
23 \( 1 + 1.45e14iT - 3.94e28T^{2} \)
29 \( 1 - 2.54e15T + 5.13e30T^{2} \)
31 \( 1 + 8.14e13T + 2.08e31T^{2} \)
37 \( 1 - 5.96e15iT - 8.55e32T^{2} \)
41 \( 1 - 2.16e16T + 7.38e33T^{2} \)
43 \( 1 + 5.16e15iT - 2.00e34T^{2} \)
47 \( 1 - 4.47e17iT - 1.30e35T^{2} \)
53 \( 1 - 2.16e17iT - 1.62e36T^{2} \)
59 \( 1 + 3.65e18T + 1.54e37T^{2} \)
61 \( 1 + 6.71e17T + 3.10e37T^{2} \)
67 \( 1 - 5.53e18iT - 2.22e38T^{2} \)
71 \( 1 - 9.93e18T + 7.52e38T^{2} \)
73 \( 1 + 4.06e19iT - 1.34e39T^{2} \)
79 \( 1 - 6.51e19T + 7.08e39T^{2} \)
83 \( 1 + 1.48e20iT - 1.99e40T^{2} \)
89 \( 1 - 2.72e20T + 8.65e40T^{2} \)
97 \( 1 - 1.44e19iT - 5.27e41T^{2} \)
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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

−10.79803559395008466551225922649, −10.16886688052460160937433939447, −9.018275803512160066886611334907, −7.73330415224779863928332906345, −6.42425479254952411419385446755, −4.80720988044480968601870222249, −4.02485580417996297048365676686, −2.75854892864085588712123007344, −1.48516634045337926359287314511, −0.44827893462329754674462908162, 0.963860607192440324154422274797, 2.19294184349310745229076614890, 3.80840304005502823770604460927, 4.92681030021659894287104589824, 6.18750665564496695701663260027, 7.11979276330192367694261804013, 8.224599713466678641948495336945, 9.330388070921166280713291852963, 10.46515927028041985633251257966, 11.98626283136203485477977330710

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