Properties

Degree 2
Conductor $ 2 \cdot 5^{2} $
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

\( d \)  =  \(2\)
\( N \)  =  \(50\)    =    \(2 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.894 + 0.447i$
motivic weight  =  \(5\)
character  :  $\chi_{50} (49, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 50,\ (\ :5/2),\ 0.894 + 0.447i)\)
\(L(3)\)  \(\approx\)  \(1.55594 - 0.367309i\)
\(L(\frac12)\)  \(\approx\)  \(1.55594 - 0.367309i\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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