Properties

Label 2-50-5.4-c21-0-31
Degree $2$
Conductor $50$
Sign $-0.447 - 0.894i$
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 − 1.76e5i·3-s − 1.04e6·4-s + 1.80e8·6-s − 1.28e9i·7-s − 1.07e9i·8-s − 2.07e10·9-s − 8.21e9·11-s + 1.85e11i·12-s + 2.70e11i·13-s + 1.32e12·14-s + 1.09e12·16-s − 1.05e13i·17-s − 2.12e13i·18-s + 2.60e13·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.72i·3-s − 0.5·4-s + 1.22·6-s − 1.72i·7-s − 0.353i·8-s − 1.98·9-s − 0.0954·11-s + 0.863i·12-s + 0.543i·13-s + 1.22·14-s + 0.250·16-s − 1.27i·17-s − 1.40i·18-s + 0.973·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 50,\ (\ :21/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.053336774\)
\(L(\frac12)\) \(\approx\) \(1.053336774\)
\(L(\frac{23}{2})\) 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 \( 1 + 1.76e5iT - 1.04e10T^{2} \)
7 \( 1 + 1.28e9iT - 5.58e17T^{2} \)
11 \( 1 + 8.21e9T + 7.40e21T^{2} \)
13 \( 1 - 2.70e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.05e13iT - 6.90e25T^{2} \)
19 \( 1 - 2.60e13T + 7.14e26T^{2} \)
23 \( 1 + 3.27e14iT - 3.94e28T^{2} \)
29 \( 1 + 2.01e15T + 5.13e30T^{2} \)
31 \( 1 + 7.82e15T + 2.08e31T^{2} \)
37 \( 1 + 2.12e16iT - 8.55e32T^{2} \)
41 \( 1 + 9.00e16T + 7.38e33T^{2} \)
43 \( 1 + 6.03e16iT - 2.00e34T^{2} \)
47 \( 1 + 3.50e17iT - 1.30e35T^{2} \)
53 \( 1 - 1.24e18iT - 1.62e36T^{2} \)
59 \( 1 - 5.56e18T + 1.54e37T^{2} \)
61 \( 1 + 5.81e18T + 3.10e37T^{2} \)
67 \( 1 + 2.21e19iT - 2.22e38T^{2} \)
71 \( 1 - 3.61e19T + 7.52e38T^{2} \)
73 \( 1 + 4.54e19iT - 1.34e39T^{2} \)
79 \( 1 - 9.81e19T + 7.08e39T^{2} \)
83 \( 1 + 4.19e19iT - 1.99e40T^{2} \)
89 \( 1 - 3.94e20T + 8.65e40T^{2} \)
97 \( 1 - 6.34e20iT - 5.27e41T^{2} \)
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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

−10.76208607150091931618929863390, −9.170463955212343286918681883747, −7.74689955491832141169472991190, −7.22155056090179950380742305802, −6.54158130154980745370157089622, −5.08283600589039106777926751039, −3.58639329643724413087734174080, −2.00137008357279076273938992959, −0.78160322340820774005063530781, −0.27184188910551005625731584678, 1.77896898558320070385124674674, 3.06416942654994681796095911359, 3.75465655855818433421290412092, 5.27418601497150723745294859086, 5.61773368608329429618857718464, 8.220014773550244143830305036744, 9.213972709575401279287554675384, 9.811849758885052567682083413560, 11.00811144669811311143159087624, 11.77740776278390691374163050826

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