Properties

Label 2-50-5.4-c21-0-30
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 + 1.74e5i·3-s − 1.04e6·4-s + 1.78e8·6-s − 9.60e8i·7-s + 1.07e9i·8-s − 2.00e10·9-s + 8.54e10·11-s − 1.83e11i·12-s − 9.74e11i·13-s − 9.83e11·14-s + 1.09e12·16-s − 1.17e13i·17-s + 2.05e13i·18-s − 1.41e13·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.70i·3-s − 0.5·4-s + 1.20·6-s − 1.28i·7-s + 0.353i·8-s − 1.91·9-s + 0.993·11-s − 0.853i·12-s − 1.96i·13-s − 0.908·14-s + 0.250·16-s − 1.40i·17-s + 1.35i·18-s − 0.528·19-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

\(L(11)\) \(\approx\) \(0.04017791394\)
\(L(\frac12)\) \(\approx\) \(0.04017791394\)
\(L(\frac{23}{2})\) 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 \( 1 - 1.74e5iT - 1.04e10T^{2} \)
7 \( 1 + 9.60e8iT - 5.58e17T^{2} \)
11 \( 1 - 8.54e10T + 7.40e21T^{2} \)
13 \( 1 + 9.74e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.17e13iT - 6.90e25T^{2} \)
19 \( 1 + 1.41e13T + 7.14e26T^{2} \)
23 \( 1 - 2.65e13iT - 3.94e28T^{2} \)
29 \( 1 + 1.45e15T + 5.13e30T^{2} \)
31 \( 1 + 7.63e15T + 2.08e31T^{2} \)
37 \( 1 - 1.09e16iT - 8.55e32T^{2} \)
41 \( 1 - 7.93e16T + 7.38e33T^{2} \)
43 \( 1 - 8.36e16iT - 2.00e34T^{2} \)
47 \( 1 - 3.57e17iT - 1.30e35T^{2} \)
53 \( 1 - 8.17e17iT - 1.62e36T^{2} \)
59 \( 1 + 8.21e17T + 1.54e37T^{2} \)
61 \( 1 - 4.53e18T + 3.10e37T^{2} \)
67 \( 1 - 8.02e18iT - 2.22e38T^{2} \)
71 \( 1 + 5.25e19T + 7.52e38T^{2} \)
73 \( 1 + 9.28e18iT - 1.34e39T^{2} \)
79 \( 1 + 1.08e19T + 7.08e39T^{2} \)
83 \( 1 - 4.79e19iT - 1.99e40T^{2} \)
89 \( 1 + 2.35e20T + 8.65e40T^{2} \)
97 \( 1 - 7.25e20iT - 5.27e41T^{2} \)
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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

−10.66578228558381331134845908727, −9.868978748087819480834728932492, −9.022542073695481795268272170257, −7.54417252636714875647257965548, −5.61854849508492548553958271154, −4.54173793836631963184298455806, −3.72913823394362250626411401724, −2.92716915608561409333530049197, −0.976518162360805630247710172154, −0.008876411050267002943601506958, 1.61350479329723304815596916179, 2.07285163365818190915722848946, 3.95005329015847195158528765237, 5.73129483122729463667874221610, 6.43046666366326586317346366585, 7.23730730452418556431994255181, 8.595362147544818708795954486365, 9.095120171580334124259617414309, 11.40985315772941131890406644234, 12.25256315263251256326183376835

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