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Properties

Label 2-50-5.4-c21-0-11
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 + 1.94e5i·3-s − 1.04e6·4-s − 1.99e8·6-s − 9.63e8i·7-s − 1.07e9i·8-s − 2.74e10·9-s + 6.10e10·11-s − 2.04e11i·12-s + 5.25e10i·13-s + 9.86e11·14-s + 1.09e12·16-s + 1.13e13i·17-s − 2.81e13i·18-s + 4.53e13·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.90i·3-s − 0.5·4-s − 1.34·6-s − 1.28i·7-s − 0.353i·8-s − 2.62·9-s + 0.710·11-s − 0.952i·12-s + 0.105i·13-s + 0.911·14-s + 0.250·16-s + 1.36i·17-s − 1.85i·18-s + 1.69·19-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

\(L(11)\) \(\approx\) \(1.882480787\)
\(L(\frac12)\) \(\approx\) \(1.882480787\)
\(L(\frac{23}{2})\) 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 \( 1 - 1.94e5iT - 1.04e10T^{2} \)
7 \( 1 + 9.63e8iT - 5.58e17T^{2} \)
11 \( 1 - 6.10e10T + 7.40e21T^{2} \)
13 \( 1 - 5.25e10iT - 2.47e23T^{2} \)
17 \( 1 - 1.13e13iT - 6.90e25T^{2} \)
19 \( 1 - 4.53e13T + 7.14e26T^{2} \)
23 \( 1 - 2.47e14iT - 3.94e28T^{2} \)
29 \( 1 - 2.58e15T + 5.13e30T^{2} \)
31 \( 1 + 1.35e15T + 2.08e31T^{2} \)
37 \( 1 - 1.35e16iT - 8.55e32T^{2} \)
41 \( 1 - 6.72e16T + 7.38e33T^{2} \)
43 \( 1 + 3.70e16iT - 2.00e34T^{2} \)
47 \( 1 + 3.17e17iT - 1.30e35T^{2} \)
53 \( 1 - 1.44e18iT - 1.62e36T^{2} \)
59 \( 1 - 1.08e18T + 1.54e37T^{2} \)
61 \( 1 + 1.59e17T + 3.10e37T^{2} \)
67 \( 1 + 8.71e18iT - 2.22e38T^{2} \)
71 \( 1 + 3.06e19T + 7.52e38T^{2} \)
73 \( 1 - 2.36e19iT - 1.34e39T^{2} \)
79 \( 1 - 9.35e19T + 7.08e39T^{2} \)
83 \( 1 + 2.00e20iT - 1.99e40T^{2} \)
89 \( 1 + 3.85e19T + 8.65e40T^{2} \)
97 \( 1 - 5.46e20iT - 5.27e41T^{2} \)
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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

−11.68614096579788702706273841717, −10.53766194634199089108870417186, −9.812361664183155257591147292372, −8.861464334424107123779851677522, −7.56807413602107299623169453603, −6.06874814747995062398886215064, −4.94809238988077000184099397883, −3.97708691260810712916228304919, −3.39887912351226729155351268830, −1.00831111670991470962563323825, 0.46893672753631565036018035023, 1.24326415165762654494283528632, 2.39036132116305903645881987980, 2.97682548156345517873023580680, 5.16567123559486270438894205788, 6.21421893882950211177726046440, 7.34273953035467541447688631624, 8.498377351967123532740545818640, 9.378191078775384167874399191674, 11.39017850972601505305373646038

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