Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{2} \cdot 5 $
Sign $-0.423 + 0.905i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.254 − 2.22i)5-s − 2.64i·7-s − 1.51i·11-s + 3.87·13-s − 3.31i·17-s + 7.08i·19-s − 4.82i·23-s + (−4.87 − 1.12i)25-s − 2.18i·29-s + 7.36·31-s + (−5.87 − 0.671i)35-s − 7.87·37-s − 8.72·41-s − 1.01·43-s − 7.08i·47-s + ⋯
L(s)  = 1  + (0.113 − 0.993i)5-s − 0.998i·7-s − 0.456i·11-s + 1.07·13-s − 0.803i·17-s + 1.62i·19-s − 1.00i·23-s + (−0.974 − 0.225i)25-s − 0.405i·29-s + 1.32·31-s + (−0.992 − 0.113i)35-s − 1.29·37-s − 1.36·41-s − 0.155·43-s − 1.03i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $-0.423 + 0.905i$
motivic weight  =  \(1\)
character  :  $\chi_{1440} (1009, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1440,\ (\ :1/2),\ -0.423 + 0.905i)\)
\(L(1)\)  \(\approx\)  \(1.535459161\)
\(L(\frac12)\)  \(\approx\)  \(1.535459161\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.254 + 2.22i)T \)
good7 \( 1 + 2.64iT - 7T^{2} \)
11 \( 1 + 1.51iT - 11T^{2} \)
13 \( 1 - 3.87T + 13T^{2} \)
17 \( 1 + 3.31iT - 17T^{2} \)
19 \( 1 - 7.08iT - 19T^{2} \)
23 \( 1 + 4.82iT - 23T^{2} \)
29 \( 1 + 2.18iT - 29T^{2} \)
31 \( 1 - 7.36T + 31T^{2} \)
37 \( 1 + 7.87T + 37T^{2} \)
41 \( 1 + 8.72T + 41T^{2} \)
43 \( 1 + 1.01T + 43T^{2} \)
47 \( 1 + 7.08iT - 47T^{2} \)
53 \( 1 + 4.50T + 53T^{2} \)
59 \( 1 - 6.79iT - 59T^{2} \)
61 \( 1 + 3.60iT - 61T^{2} \)
67 \( 1 - 1.01T + 67T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 + 15.5iT - 73T^{2} \)
79 \( 1 + 7.36T + 79T^{2} \)
83 \( 1 - 7.74T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−9.180513667354531415699426001228, −8.393844923273724910564295440307, −7.911177608819110983359297392566, −6.75786698426310170069891003407, −5.98571442105713457459918030359, −5.02677295304508982351845960538, −4.14661494412211771573830890763, −3.35456174967073238826948458896, −1.69220754793623285507704848024, −0.63387276770046806625883227740, 1.68848372232822419363629281888, 2.77057768298575599662989351449, 3.58491394484793003804569950634, 4.83587466011782058375013325307, 5.80530347700276820238898848802, 6.51786876688558097952813339302, 7.22120526023901260487492168697, 8.314328801644740341250531006040, 8.930991393144291870459046366458, 9.810226818428075261460677532192

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