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Properties

Label 2-50-5.4-c21-0-18
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 − 5.93e4i·3-s − 1.04e6·4-s + 6.07e7·6-s + 1.42e9i·7-s − 1.07e9i·8-s + 6.94e9·9-s − 1.06e11·11-s + 6.21e10i·12-s + 1.50e11i·13-s − 1.46e12·14-s + 1.09e12·16-s − 1.12e13i·17-s + 7.10e12i·18-s − 1.10e13·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.579i·3-s − 0.5·4-s + 0.410·6-s + 1.90i·7-s − 0.353i·8-s + 0.663·9-s − 1.24·11-s + 0.289i·12-s + 0.302i·13-s − 1.35·14-s + 0.250·16-s − 1.34i·17-s + 0.469i·18-s − 0.412·19-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

\(L(11)\) \(\approx\) \(1.134169518\)
\(L(\frac12)\) \(\approx\) \(1.134169518\)
\(L(\frac{23}{2})\) 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 \( 1 + 5.93e4iT - 1.04e10T^{2} \)
7 \( 1 - 1.42e9iT - 5.58e17T^{2} \)
11 \( 1 + 1.06e11T + 7.40e21T^{2} \)
13 \( 1 - 1.50e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.12e13iT - 6.90e25T^{2} \)
19 \( 1 + 1.10e13T + 7.14e26T^{2} \)
23 \( 1 + 1.29e14iT - 3.94e28T^{2} \)
29 \( 1 + 2.38e15T + 5.13e30T^{2} \)
31 \( 1 + 8.78e14T + 2.08e31T^{2} \)
37 \( 1 - 3.11e16iT - 8.55e32T^{2} \)
41 \( 1 + 2.46e16T + 7.38e33T^{2} \)
43 \( 1 - 1.33e17iT - 2.00e34T^{2} \)
47 \( 1 + 1.92e17iT - 1.30e35T^{2} \)
53 \( 1 - 5.94e17iT - 1.62e36T^{2} \)
59 \( 1 - 2.95e18T + 1.54e37T^{2} \)
61 \( 1 - 7.98e18T + 3.10e37T^{2} \)
67 \( 1 - 4.83e18iT - 2.22e38T^{2} \)
71 \( 1 - 8.84e18T + 7.52e38T^{2} \)
73 \( 1 + 3.66e19iT - 1.34e39T^{2} \)
79 \( 1 + 3.38e19T + 7.08e39T^{2} \)
83 \( 1 + 2.04e20iT - 1.99e40T^{2} \)
89 \( 1 - 4.10e19T + 8.65e40T^{2} \)
97 \( 1 + 7.27e20iT - 5.27e41T^{2} \)
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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

−11.58853654263874544021090680266, −9.886596085472561051499759243879, −8.823304205727543437255640842190, −7.84732627649111512030196500606, −6.73040802983817415100185122221, −5.62512160876956651533427933687, −4.75681416305888113936882787643, −2.82967795509440754675334941170, −1.92488484103355216257403409450, −0.28836754703506410747344116589, 0.78606570688066628316651994737, 1.92073137547734546886911275297, 3.60778714657131260154139789276, 4.09982424601330285745144594225, 5.32638353637116988902662087435, 7.12176441442602752759265731693, 8.084256301051287213614922015362, 9.728506530426118375115913930014, 10.51061021540967764248500975055, 10.93630812341661079768839303231

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