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Properties

Label 2-50-5.4-c27-0-17
Degree $2$
Conductor $50$
Sign $-0.894 - 0.447i$
Analytic cond. $230.927$
Root an. cond. $15.1963$
Motivic weight $27$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.19e3i·2-s + 5.48e6i·3-s − 6.71e7·4-s − 4.49e10·6-s − 8.15e10i·7-s − 5.49e11i·8-s − 2.25e13·9-s + 1.33e13·11-s − 3.68e14i·12-s − 5.64e14i·13-s + 6.68e14·14-s + 4.50e15·16-s − 1.28e16i·17-s − 1.84e17i·18-s − 3.07e17·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.98i·3-s − 0.5·4-s − 1.40·6-s − 0.318i·7-s − 0.353i·8-s − 2.95·9-s + 0.116·11-s − 0.993i·12-s − 0.517i·13-s + 0.225·14-s + 0.250·16-s − 0.315i·17-s − 2.08i·18-s − 1.67·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(230.927\)
Root analytic conductor: \(15.1963\)
Motivic weight: \(27\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :27/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(14)\) \(\approx\) \(1.431346617\)
\(L(\frac12)\) \(\approx\) \(1.431346617\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8.19e3iT \)
5 \( 1 \)
good3 \( 1 - 5.48e6iT - 7.62e12T^{2} \)
7 \( 1 + 8.15e10iT - 6.57e22T^{2} \)
11 \( 1 - 1.33e13T + 1.31e28T^{2} \)
13 \( 1 + 5.64e14iT - 1.19e30T^{2} \)
17 \( 1 + 1.28e16iT - 1.66e33T^{2} \)
19 \( 1 + 3.07e17T + 3.36e34T^{2} \)
23 \( 1 - 1.99e18iT - 5.84e36T^{2} \)
29 \( 1 - 8.80e19T + 3.05e39T^{2} \)
31 \( 1 - 2.22e20T + 1.84e40T^{2} \)
37 \( 1 + 1.61e21iT - 2.19e42T^{2} \)
41 \( 1 + 9.68e21T + 3.50e43T^{2} \)
43 \( 1 + 9.33e21iT - 1.26e44T^{2} \)
47 \( 1 - 1.20e22iT - 1.40e45T^{2} \)
53 \( 1 - 8.75e21iT - 3.59e46T^{2} \)
59 \( 1 + 3.03e23T + 6.50e47T^{2} \)
61 \( 1 - 1.15e24T + 1.59e48T^{2} \)
67 \( 1 + 1.86e24iT - 2.01e49T^{2} \)
71 \( 1 + 8.93e24T + 9.63e49T^{2} \)
73 \( 1 - 2.18e25iT - 2.04e50T^{2} \)
79 \( 1 + 1.05e25T + 1.72e51T^{2} \)
83 \( 1 + 2.25e25iT - 6.53e51T^{2} \)
89 \( 1 - 1.78e26T + 4.30e52T^{2} \)
97 \( 1 - 8.87e26iT - 4.39e53T^{2} \)
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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

−10.60218380176521455591236808650, −10.06538293571587494764278245767, −8.924367600055644875522832160643, −8.192648595310860244475928608454, −6.47794570480739075444679766759, −5.40158134819989508968392064734, −4.53404753323030807465923418951, −3.78511207564267139353912027633, −2.67550122561210648813326303077, −0.51720399986407248519638729842, 0.47016631638348800598066543150, 1.38675870211078980221414592935, 2.21324781918057666124416571049, 2.96103703145723988502440073305, 4.65121671499035767810438110000, 6.19567485907404402109159007370, 6.72903488593664130350733688201, 8.259069812383715568312615536767, 8.636124076136965340608240468523, 10.42375274912640410577254335383

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