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Properties

Label 2-50-5.4-c11-0-3
Degree $2$
Conductor $50$
Sign $0.894 + 0.447i$
Analytic cond. $38.4171$
Root an. cond. $6.19815$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 32i·2-s − 738i·3-s − 1.02e3·4-s − 2.36e4·6-s + 2.55e4i·7-s + 3.27e4i·8-s − 3.67e5·9-s + 7.69e5·11-s + 7.55e5i·12-s + 9.18e5i·13-s + 8.18e5·14-s + 1.04e6·16-s + 1.03e7i·17-s + 1.17e7i·18-s + 5.52e6·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.75i·3-s − 0.5·4-s − 1.23·6-s + 0.575i·7-s + 0.353i·8-s − 2.07·9-s + 1.43·11-s + 0.876i·12-s + 0.686i·13-s + 0.406·14-s + 0.250·16-s + 1.76i·17-s + 1.46i·18-s + 0.511·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(38.4171\)
Root analytic conductor: \(6.19815\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :11/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.35726 - 0.320405i\)
\(L(\frac12)\) \(\approx\) \(1.35726 - 0.320405i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 32iT \)
5 \( 1 \)
good3 \( 1 + 738iT - 1.77e5T^{2} \)
7 \( 1 - 2.55e4iT - 1.97e9T^{2} \)
11 \( 1 - 7.69e5T + 2.85e11T^{2} \)
13 \( 1 - 9.18e5iT - 1.79e12T^{2} \)
17 \( 1 - 1.03e7iT - 3.42e13T^{2} \)
19 \( 1 - 5.52e6T + 1.16e14T^{2} \)
23 \( 1 - 3.99e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.52e7T + 1.22e16T^{2} \)
31 \( 1 + 2.41e8T + 2.54e16T^{2} \)
37 \( 1 + 2.57e7iT - 1.77e17T^{2} \)
41 \( 1 + 1.21e9T + 5.50e17T^{2} \)
43 \( 1 - 6.83e8iT - 9.29e17T^{2} \)
47 \( 1 - 1.53e9iT - 2.47e18T^{2} \)
53 \( 1 + 3.57e9iT - 9.26e18T^{2} \)
59 \( 1 - 1.06e9T + 3.01e19T^{2} \)
61 \( 1 + 2.09e9T + 4.35e19T^{2} \)
67 \( 1 + 1.46e9iT - 1.22e20T^{2} \)
71 \( 1 - 9.66e9T + 2.31e20T^{2} \)
73 \( 1 - 5.60e9iT - 3.13e20T^{2} \)
79 \( 1 + 5.02e9T + 7.47e20T^{2} \)
83 \( 1 - 3.84e10iT - 1.28e21T^{2} \)
89 \( 1 + 3.55e10T + 2.77e21T^{2} \)
97 \( 1 - 1.05e10iT - 7.15e21T^{2} \)
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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

−12.84094791361212351450409990256, −12.02863045394149572243413709304, −11.28906801214359353324378329024, −9.316861489261240639280842434663, −8.273258273009780136152880398762, −6.88426667724267760994203351060, −5.79310703056785406080803712480, −3.59670398752803617786448697997, −1.89208478088729924246311446951, −1.32208407137078669735297116324, 0.43359200484065664203038252802, 3.32029807235470095507372730290, 4.37887344370982526321822840442, 5.42164230827264856408123896171, 7.02146431354943971697093230308, 8.747855133869243472275023092923, 9.577637996368804841097005670537, 10.59633224585332346227590844853, 11.85579964257498970280137093382, 13.82769493344400355724605535809

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