Properties

Degree $2$
Conductor $50$
Sign $0.894 - 0.447i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s + 6i·3-s − 16·4-s + 24·6-s + 118i·7-s + 64i·8-s + 207·9-s + 192·11-s − 96i·12-s + 1.10e3i·13-s + 472·14-s + 256·16-s − 762i·17-s − 828i·18-s + 2.74e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.384i·3-s − 0.5·4-s + 0.272·6-s + 0.910i·7-s + 0.353i·8-s + 0.851·9-s + 0.478·11-s − 0.192i·12-s + 1.81i·13-s + 0.643·14-s + 0.250·16-s − 0.639i·17-s − 0.602i·18-s + 1.74·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

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

Particular Values

\(L(3)\) \(\approx\) \(1.55594 + 0.367309i\)
\(L(\frac12)\) \(\approx\) \(1.55594 + 0.367309i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
5 \( 1 \)
good3 \( 1 - 6iT - 243T^{2} \)
7 \( 1 - 118iT - 1.68e4T^{2} \)
11 \( 1 - 192T + 1.61e5T^{2} \)
13 \( 1 - 1.10e3iT - 3.71e5T^{2} \)
17 \( 1 + 762iT - 1.41e6T^{2} \)
19 \( 1 - 2.74e3T + 2.47e6T^{2} \)
23 \( 1 - 1.56e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.91e3T + 2.05e7T^{2} \)
31 \( 1 + 6.86e3T + 2.86e7T^{2} \)
37 \( 1 - 5.51e3iT - 6.93e7T^{2} \)
41 \( 1 + 378T + 1.15e8T^{2} \)
43 \( 1 + 2.43e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.31e4iT - 2.29e8T^{2} \)
53 \( 1 + 9.17e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.49e4T + 7.14e8T^{2} \)
61 \( 1 + 9.83e3T + 8.44e8T^{2} \)
67 \( 1 + 3.37e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.02e4T + 1.80e9T^{2} \)
73 \( 1 - 2.19e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.52e3T + 3.07e9T^{2} \)
83 \( 1 + 1.09e5iT - 3.93e9T^{2} \)
89 \( 1 + 3.84e4T + 5.58e9T^{2} \)
97 \( 1 - 1.91e3iT - 8.58e9T^{2} \)
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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

−14.52155241674561830901334391163, −13.41927511585250737252868425132, −12.01541310121538520557986715212, −11.36419583327179316600783836124, −9.625684455243939828521651815085, −9.151728792740791927997915396845, −7.14875273808200615221893091120, −5.22664728751887930531651339036, −3.73533592306789052498690499197, −1.76136875721461425397016243766, 0.931167732643556534110499541698, 3.78151259459389169613567621256, 5.54341723371841183500262925621, 7.12274080070210985605437915698, 7.88267787439678049569286541262, 9.613370307717633466875612957780, 10.75092115404869162332093097767, 12.56495785981300643267951314590, 13.33485274730131380284671185393, 14.51085397284922101574563877189

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