Properties

Label 2-50-5.4-c21-0-28
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 − 7.16e4i·3-s − 1.04e6·4-s − 7.33e7·6-s − 8.53e8i·7-s + 1.07e9i·8-s + 5.33e9·9-s + 8.67e10·11-s + 7.50e10i·12-s + 8.95e11i·13-s − 8.73e11·14-s + 1.09e12·16-s + 3.25e12i·17-s − 5.46e12i·18-s − 2.30e13·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.700i·3-s − 0.5·4-s − 0.495·6-s − 1.14i·7-s + 0.353i·8-s + 0.509·9-s + 1.00·11-s + 0.350i·12-s + 1.80i·13-s − 0.807·14-s + 0.250·16-s + 0.391i·17-s − 0.360i·18-s − 0.861·19-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

\(L(11)\) \(\approx\) \(1.482271877\)
\(L(\frac12)\) \(\approx\) \(1.482271877\)
\(L(\frac{23}{2})\) 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 \( 1 + 7.16e4iT - 1.04e10T^{2} \)
7 \( 1 + 8.53e8iT - 5.58e17T^{2} \)
11 \( 1 - 8.67e10T + 7.40e21T^{2} \)
13 \( 1 - 8.95e11iT - 2.47e23T^{2} \)
17 \( 1 - 3.25e12iT - 6.90e25T^{2} \)
19 \( 1 + 2.30e13T + 7.14e26T^{2} \)
23 \( 1 + 1.46e14iT - 3.94e28T^{2} \)
29 \( 1 - 7.34e14T + 5.13e30T^{2} \)
31 \( 1 + 3.14e15T + 2.08e31T^{2} \)
37 \( 1 + 1.29e16iT - 8.55e32T^{2} \)
41 \( 1 - 4.57e16T + 7.38e33T^{2} \)
43 \( 1 - 2.40e16iT - 2.00e34T^{2} \)
47 \( 1 + 4.49e17iT - 1.30e35T^{2} \)
53 \( 1 + 2.06e18iT - 1.62e36T^{2} \)
59 \( 1 - 3.78e18T + 1.54e37T^{2} \)
61 \( 1 + 7.61e18T + 3.10e37T^{2} \)
67 \( 1 + 1.87e19iT - 2.22e38T^{2} \)
71 \( 1 + 4.52e18T + 7.52e38T^{2} \)
73 \( 1 - 2.55e19iT - 1.34e39T^{2} \)
79 \( 1 + 9.93e19T + 7.08e39T^{2} \)
83 \( 1 + 2.95e18iT - 1.99e40T^{2} \)
89 \( 1 + 1.18e20T + 8.65e40T^{2} \)
97 \( 1 + 5.69e20iT - 5.27e41T^{2} \)
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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

−10.86711124740368675642674380152, −9.753766483359357318616796002707, −8.607713382864770684644181936703, −7.14237640540903218473166381959, −6.47733256628916341646336513870, −4.38853788091406564196720895965, −3.88761926788114734625837394582, −2.04072672457927557809682042989, −1.38824926949537898844379134907, −0.31068542135626789157765065673, 1.20447549963689638552495716423, 2.86879987016955764802847820089, 4.08442027957201575427829574279, 5.24290729183518708497856882071, 6.11879974996111301397018440080, 7.51898062573216813335713698643, 8.745507369355611425893436275733, 9.564958831726811156665950002592, 10.70206063683238196835383542975, 12.17353245203254450024984872312

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