Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} $
Sign $0.447 - 0.894i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22i·3-s + 218i·7-s − 241·9-s − 480·11-s + 622i·13-s + 186i·17-s + 1.20e3·19-s + 4.79e3·21-s + 3.18e3i·23-s − 44i·27-s − 5.52e3·29-s + 9.35e3·31-s + 1.05e4i·33-s + 5.61e3i·37-s + 1.36e4·39-s + ⋯
L(s)  = 1  − 1.41i·3-s + 1.68i·7-s − 0.991·9-s − 1.19·11-s + 1.02i·13-s + 0.156i·17-s + 0.765·19-s + 2.37·21-s + 1.25i·23-s − 0.0116i·27-s − 1.22·29-s + 1.74·31-s + 1.68i·33-s + 0.674i·37-s + 1.44·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100\)    =    \(2^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.447 - 0.894i$
motivic weight  =  \(5\)
character  :  $\chi_{100} (49, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 100,\ (\ :5/2),\ 0.447 - 0.894i)\)
\(L(3)\)  \(\approx\)  \(0.948069 + 0.585938i\)
\(L(\frac12)\)  \(\approx\)  \(0.948069 + 0.585938i\)
\(L(\frac{7}{2})\)   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 \( 1 \)
5 \( 1 \)
good3 \( 1 + 22iT - 243T^{2} \)
7 \( 1 - 218iT - 1.68e4T^{2} \)
11 \( 1 + 480T + 1.61e5T^{2} \)
13 \( 1 - 622iT - 3.71e5T^{2} \)
17 \( 1 - 186iT - 1.41e6T^{2} \)
19 \( 1 - 1.20e3T + 2.47e6T^{2} \)
23 \( 1 - 3.18e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.52e3T + 2.05e7T^{2} \)
31 \( 1 - 9.35e3T + 2.86e7T^{2} \)
37 \( 1 - 5.61e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.43e4T + 1.15e8T^{2} \)
43 \( 1 - 370iT - 1.47e8T^{2} \)
47 \( 1 - 1.61e4iT - 2.29e8T^{2} \)
53 \( 1 - 4.37e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.17e4T + 7.14e8T^{2} \)
61 \( 1 - 1.32e4T + 8.44e8T^{2} \)
67 \( 1 + 1.15e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.95e4T + 1.80e9T^{2} \)
73 \( 1 + 3.36e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.12e4T + 3.07e9T^{2} \)
83 \( 1 - 3.84e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.19e5T + 5.58e9T^{2} \)
97 \( 1 - 9.46e4iT - 8.58e9T^{2} \)
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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

−13.06765048208727941147586477003, −12.06029969465593194594557080627, −11.51420899461360142956913198021, −9.673403851168556882213035860148, −8.486876319914660903375291367264, −7.54330958615218194979753127202, −6.28726574150430824333206229539, −5.25920091769356198073809954037, −2.76210968078179921982665480749, −1.66100540952640138243097922729, 0.42922929739508661423373705015, 3.15968668002452611948618399455, 4.32708726719985766938909321496, 5.34886129743555918672349690512, 7.21395051296656781889049430941, 8.351436312350168455783687219368, 10.10016586258612539175154861715, 10.21215189734531821802357364648, 11.23067453634351279752107739075, 12.94413892549526068153423632625

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