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Properties

Label 2-50-5.4-c25-0-11
Degree $2$
Conductor $50$
Sign $0.894 - 0.447i$
Analytic cond. $197.998$
Root an. cond. $14.0711$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.09e3i·2-s + 1.62e5i·3-s − 1.67e7·4-s + 6.67e8·6-s + 1.76e10i·7-s + 6.87e10i·8-s + 8.20e11·9-s − 1.12e13·11-s − 2.73e12i·12-s + 4.24e12i·13-s + 7.20e13·14-s + 2.81e14·16-s + 1.13e15i·17-s − 3.36e15i·18-s − 2.98e15·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.176i·3-s − 0.5·4-s + 0.125·6-s + 0.480i·7-s + 0.353i·8-s + 0.968·9-s − 1.07·11-s − 0.0884i·12-s + 0.0505i·13-s + 0.339·14-s + 0.250·16-s + 0.473i·17-s − 0.684i·18-s − 0.309·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(197.998\)
Root analytic conductor: \(14.0711\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :25/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(13)\) \(\approx\) \(1.695584857\)
\(L(\frac12)\) \(\approx\) \(1.695584857\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4.09e3iT \)
5 \( 1 \)
good3 \( 1 - 1.62e5iT - 8.47e11T^{2} \)
7 \( 1 - 1.76e10iT - 1.34e21T^{2} \)
11 \( 1 + 1.12e13T + 1.08e26T^{2} \)
13 \( 1 - 4.24e12iT - 7.05e27T^{2} \)
17 \( 1 - 1.13e15iT - 5.77e30T^{2} \)
19 \( 1 + 2.98e15T + 9.30e31T^{2} \)
23 \( 1 + 1.22e17iT - 1.10e34T^{2} \)
29 \( 1 - 2.12e18T + 3.63e36T^{2} \)
31 \( 1 - 4.22e18T + 1.92e37T^{2} \)
37 \( 1 + 2.41e19iT - 1.60e39T^{2} \)
41 \( 1 + 1.61e20T + 2.08e40T^{2} \)
43 \( 1 + 3.11e20iT - 6.86e40T^{2} \)
47 \( 1 - 1.22e21iT - 6.34e41T^{2} \)
53 \( 1 + 4.78e20iT - 1.27e43T^{2} \)
59 \( 1 + 1.21e22T + 1.86e44T^{2} \)
61 \( 1 + 2.03e22T + 4.29e44T^{2} \)
67 \( 1 + 6.15e22iT - 4.48e45T^{2} \)
71 \( 1 - 1.34e23T + 1.91e46T^{2} \)
73 \( 1 - 2.23e23iT - 3.82e46T^{2} \)
79 \( 1 + 3.82e23T + 2.75e47T^{2} \)
83 \( 1 + 7.71e23iT - 9.48e47T^{2} \)
89 \( 1 - 8.40e23T + 5.42e48T^{2} \)
97 \( 1 - 6.19e24iT - 4.66e49T^{2} \)
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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

−10.69153528165787971105627668688, −10.15417686533940605923419673761, −8.887756914342835971782710552495, −7.86878182171569829002187791916, −6.43271720711638530179298423396, −5.06925420225393419449581586950, −4.20696234952478921706195446948, −2.89091712956766744060535347024, −1.99708276503993726520539609957, −0.78418962322735280677430921680, 0.40398329640337315461319682551, 1.49527397258355509241562272706, 2.97633327446984490235084131450, 4.31385626968626866628725584132, 5.20602737546580880107925699312, 6.54124816270118636361090111646, 7.42729367330955195762598628777, 8.276248078726072387486559006916, 9.726858456026688443134010276292, 10.50689703299630834892066667024

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