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Properties

Label 2-50-5.4-c21-0-3
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 + 4.79e4i·3-s − 1.04e6·4-s − 4.90e7·6-s − 6.67e8i·7-s − 1.07e9i·8-s + 8.16e9·9-s − 1.26e11·11-s − 5.02e10i·12-s + 9.12e11i·13-s + 6.83e11·14-s + 1.09e12·16-s − 8.60e12i·17-s + 8.35e12i·18-s + 6.93e12·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.468i·3-s − 0.5·4-s − 0.331·6-s − 0.893i·7-s − 0.353i·8-s + 0.780·9-s − 1.47·11-s − 0.234i·12-s + 1.83i·13-s + 0.631·14-s + 0.250·16-s − 1.03i·17-s + 0.551i·18-s + 0.259·19-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

\(L(11)\) \(\approx\) \(0.6518505926\)
\(L(\frac12)\) \(\approx\) \(0.6518505926\)
\(L(\frac{23}{2})\) 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 \( 1 - 4.79e4iT - 1.04e10T^{2} \)
7 \( 1 + 6.67e8iT - 5.58e17T^{2} \)
11 \( 1 + 1.26e11T + 7.40e21T^{2} \)
13 \( 1 - 9.12e11iT - 2.47e23T^{2} \)
17 \( 1 + 8.60e12iT - 6.90e25T^{2} \)
19 \( 1 - 6.93e12T + 7.14e26T^{2} \)
23 \( 1 + 3.30e14iT - 3.94e28T^{2} \)
29 \( 1 - 3.90e15T + 5.13e30T^{2} \)
31 \( 1 - 3.32e15T + 2.08e31T^{2} \)
37 \( 1 - 4.33e16iT - 8.55e32T^{2} \)
41 \( 1 + 9.56e16T + 7.38e33T^{2} \)
43 \( 1 + 7.81e16iT - 2.00e34T^{2} \)
47 \( 1 - 2.03e17iT - 1.30e35T^{2} \)
53 \( 1 - 1.42e18iT - 1.62e36T^{2} \)
59 \( 1 + 2.28e18T + 1.54e37T^{2} \)
61 \( 1 + 5.32e18T + 3.10e37T^{2} \)
67 \( 1 - 1.31e19iT - 2.22e38T^{2} \)
71 \( 1 + 2.98e19T + 7.52e38T^{2} \)
73 \( 1 - 1.37e18iT - 1.34e39T^{2} \)
79 \( 1 + 1.74e19T + 7.08e39T^{2} \)
83 \( 1 - 3.75e19iT - 1.99e40T^{2} \)
89 \( 1 + 2.43e20T + 8.65e40T^{2} \)
97 \( 1 - 1.80e20iT - 5.27e41T^{2} \)
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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

−12.09660433003739205630031575721, −10.56250413421860032654403605073, −9.857530635919108203776160903299, −8.564248077451609442806864292319, −7.31040638030936086077765291846, −6.56350959882920939534522970272, −4.74810983611518853019230784575, −4.43335434554314663760205284043, −2.78200163797293416087360571467, −1.12818701742824319067264261484, 0.14147877934534836486653220296, 1.30605383770036417843164441849, 2.42762114164923525205461769265, 3.34037971709274239522858428802, 4.99743225430063739331878990306, 5.87460520655683148254581752535, 7.63621171318291900881168115488, 8.377688139741328559725422142690, 9.950445786858392836858713688328, 10.62419085883714195499801277892

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