Properties

Label 2-50-5.4-c21-0-13
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 − 1.56e5i·3-s − 1.04e6·4-s − 1.60e8·6-s − 4.70e7i·7-s + 1.07e9i·8-s − 1.40e10·9-s − 7.06e10·11-s + 1.64e11i·12-s + 4.22e11i·13-s − 4.81e10·14-s + 1.09e12·16-s + 9.39e12i·17-s + 1.44e13i·18-s + 1.75e13·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.53i·3-s − 0.5·4-s − 1.08·6-s − 0.0629i·7-s + 0.353i·8-s − 1.34·9-s − 0.821·11-s + 0.766i·12-s + 0.849i·13-s − 0.0445·14-s + 0.250·16-s + 1.13i·17-s + 0.952i·18-s + 0.657·19-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

\(L(11)\) \(\approx\) \(1.575002939\)
\(L(\frac12)\) \(\approx\) \(1.575002939\)
\(L(\frac{23}{2})\) 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 \( 1 + 1.56e5iT - 1.04e10T^{2} \)
7 \( 1 + 4.70e7iT - 5.58e17T^{2} \)
11 \( 1 + 7.06e10T + 7.40e21T^{2} \)
13 \( 1 - 4.22e11iT - 2.47e23T^{2} \)
17 \( 1 - 9.39e12iT - 6.90e25T^{2} \)
19 \( 1 - 1.75e13T + 7.14e26T^{2} \)
23 \( 1 - 2.12e13iT - 3.94e28T^{2} \)
29 \( 1 + 3.71e15T + 5.13e30T^{2} \)
31 \( 1 - 4.95e15T + 2.08e31T^{2} \)
37 \( 1 - 2.52e15iT - 8.55e32T^{2} \)
41 \( 1 + 1.20e17T + 7.38e33T^{2} \)
43 \( 1 - 1.31e17iT - 2.00e34T^{2} \)
47 \( 1 + 6.43e17iT - 1.30e35T^{2} \)
53 \( 1 - 3.87e17iT - 1.62e36T^{2} \)
59 \( 1 - 6.13e18T + 1.54e37T^{2} \)
61 \( 1 - 9.11e18T + 3.10e37T^{2} \)
67 \( 1 + 1.33e19iT - 2.22e38T^{2} \)
71 \( 1 - 4.55e18T + 7.52e38T^{2} \)
73 \( 1 + 3.93e19iT - 1.34e39T^{2} \)
79 \( 1 - 7.55e19T + 7.08e39T^{2} \)
83 \( 1 - 3.70e19iT - 1.99e40T^{2} \)
89 \( 1 + 1.83e19T + 8.65e40T^{2} \)
97 \( 1 - 1.19e21iT - 5.27e41T^{2} \)
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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

−11.35258517378095948546015355793, −10.04929479411154026131274278941, −8.601701213762483182346969460160, −7.65920084544140778524563838841, −6.56662531586038486947598132997, −5.32189511877167713279926878713, −3.70573830806811808207781921006, −2.33837254708280601196728416524, −1.63369194100483041403941220081, −0.57626109160075253622732588977, 0.55118660905697528899142308427, 2.74798979927742836544401875214, 3.80610958479561450419159157655, 5.02834757518804342221631973943, 5.55740207088616064534919449950, 7.28091526292658143947992645504, 8.471252120983872158471629517579, 9.591951357591014722989621259461, 10.30112195479940870925092039018, 11.48727165012026016805684323414

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