Properties

Label 2-50-5.4-c21-0-19
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 + 1.22e4i·3-s − 1.04e6·4-s + 1.25e7·6-s − 9.06e8i·7-s + 1.07e9i·8-s + 1.03e10·9-s + 8.75e10·11-s − 1.28e10i·12-s + 3.78e10i·13-s − 9.27e11·14-s + 1.09e12·16-s − 1.03e13i·17-s − 1.05e13i·18-s + 4.47e13·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.119i·3-s − 0.5·4-s + 0.0845·6-s − 1.21i·7-s + 0.353i·8-s + 0.985·9-s + 1.01·11-s − 0.0598i·12-s + 0.0761i·13-s − 0.857·14-s + 0.250·16-s − 1.24i·17-s − 0.696i·18-s + 1.67·19-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

\(L(11)\) \(\approx\) \(2.884794412\)
\(L(\frac12)\) \(\approx\) \(2.884794412\)
\(L(\frac{23}{2})\) 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 \( 1 - 1.22e4iT - 1.04e10T^{2} \)
7 \( 1 + 9.06e8iT - 5.58e17T^{2} \)
11 \( 1 - 8.75e10T + 7.40e21T^{2} \)
13 \( 1 - 3.78e10iT - 2.47e23T^{2} \)
17 \( 1 + 1.03e13iT - 6.90e25T^{2} \)
19 \( 1 - 4.47e13T + 7.14e26T^{2} \)
23 \( 1 - 2.93e14iT - 3.94e28T^{2} \)
29 \( 1 - 4.56e14T + 5.13e30T^{2} \)
31 \( 1 - 5.31e15T + 2.08e31T^{2} \)
37 \( 1 - 2.33e16iT - 8.55e32T^{2} \)
41 \( 1 + 7.54e16T + 7.38e33T^{2} \)
43 \( 1 - 1.59e17iT - 2.00e34T^{2} \)
47 \( 1 + 5.36e15iT - 1.30e35T^{2} \)
53 \( 1 - 1.47e18iT - 1.62e36T^{2} \)
59 \( 1 - 2.78e18T + 1.54e37T^{2} \)
61 \( 1 + 5.95e18T + 3.10e37T^{2} \)
67 \( 1 + 3.15e18iT - 2.22e38T^{2} \)
71 \( 1 - 3.60e19T + 7.52e38T^{2} \)
73 \( 1 - 5.16e19iT - 1.34e39T^{2} \)
79 \( 1 - 1.21e20T + 7.08e39T^{2} \)
83 \( 1 - 1.00e20iT - 1.99e40T^{2} \)
89 \( 1 + 2.72e20T + 8.65e40T^{2} \)
97 \( 1 - 5.06e20iT - 5.27e41T^{2} \)
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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

−11.27470292939658216690594938226, −9.934759889893036225192989083435, −9.468177724820047701162893413692, −7.69308438788506065430631813509, −6.81550959931269964331125970830, −5.01154526857258035796551063765, −4.03613563694413399358604571031, −3.11038169263204386075934228238, −1.33824683750705322833931216714, −0.913669874330589723364347802105, 0.816647748189281839926973140228, 2.00370898954655685192423903306, 3.57219287206846856339653282093, 4.80019834530534787569066858793, 5.99927986966738806376465771220, 6.88400869378908868044409306098, 8.201059230233311228849501625223, 9.139399784424586789883114110421, 10.21257277718661739502692596469, 11.90376220522992904895281027966

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