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Properties

Label 2-50-5.4-c27-0-22
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 − 1.86e5i·3-s − 6.71e7·4-s − 1.53e9·6-s + 1.52e11i·7-s + 5.49e11i·8-s + 7.59e12·9-s + 5.20e13·11-s + 1.25e13i·12-s + 9.66e14i·13-s + 1.24e15·14-s + 4.50e15·16-s − 8.84e15i·17-s − 6.21e16i·18-s + 4.96e16·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.0676i·3-s − 0.5·4-s − 0.0478·6-s + 0.593i·7-s + 0.353i·8-s + 0.995·9-s + 0.454·11-s + 0.0338i·12-s + 0.885i·13-s + 0.419·14-s + 0.250·16-s − 0.216i·17-s − 0.703i·18-s + 0.270·19-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

\(L(14)\) \(\approx\) \(2.545109101\)
\(L(\frac12)\) \(\approx\) \(2.545109101\)
\(L(\frac{29}{2})\) 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 \( 1 + 1.86e5iT - 7.62e12T^{2} \)
7 \( 1 - 1.52e11iT - 6.57e22T^{2} \)
11 \( 1 - 5.20e13T + 1.31e28T^{2} \)
13 \( 1 - 9.66e14iT - 1.19e30T^{2} \)
17 \( 1 + 8.84e15iT - 1.66e33T^{2} \)
19 \( 1 - 4.96e16T + 3.36e34T^{2} \)
23 \( 1 + 2.39e18iT - 5.84e36T^{2} \)
29 \( 1 + 5.37e18T + 3.05e39T^{2} \)
31 \( 1 - 1.60e20T + 1.84e40T^{2} \)
37 \( 1 + 2.24e21iT - 2.19e42T^{2} \)
41 \( 1 + 4.36e21T + 3.50e43T^{2} \)
43 \( 1 - 5.19e21iT - 1.26e44T^{2} \)
47 \( 1 - 1.42e21iT - 1.40e45T^{2} \)
53 \( 1 - 2.24e23iT - 3.59e46T^{2} \)
59 \( 1 - 1.17e24T + 6.50e47T^{2} \)
61 \( 1 + 2.05e24T + 1.59e48T^{2} \)
67 \( 1 - 3.81e24iT - 2.01e49T^{2} \)
71 \( 1 + 3.12e24T + 9.63e49T^{2} \)
73 \( 1 + 1.49e25iT - 2.04e50T^{2} \)
79 \( 1 + 1.41e25T + 1.72e51T^{2} \)
83 \( 1 - 7.95e24iT - 6.53e51T^{2} \)
89 \( 1 - 1.05e26T + 4.30e52T^{2} \)
97 \( 1 + 2.67e26iT - 4.39e53T^{2} \)
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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

−10.55047968840297568024618567723, −9.526999953125402251344945261447, −8.692908545375590944612611224099, −7.29763086716117004727119437166, −6.17026595428560803421086518842, −4.76199019410908207570161003435, −3.95270637711104094297180030933, −2.62792679848016432709684731730, −1.71152444693500540365509830926, −0.73386809447715095219945073173, 0.64504948909675018386374416495, 1.52845991585861338202274490054, 3.26456531412183853247306492901, 4.23215268641315321814035173197, 5.25946188942792238072238235237, 6.50955885932008445419870777265, 7.37851650827516733766128751731, 8.326429903701488192588645344171, 9.684102055848988218055114486895, 10.39435869605031792228961833784

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