Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 16·4-s + 25·5-s − 22·7-s + 64·8-s + 100·10-s + 768·11-s − 46·13-s − 88·14-s + 256·16-s − 378·17-s + 1.10e3·19-s + 400·20-s + 3.07e3·22-s + 1.98e3·23-s + 625·25-s − 184·26-s − 352·28-s + 5.61e3·29-s − 3.98e3·31-s + 1.02e3·32-s − 1.51e3·34-s − 550·35-s − 142·37-s + 4.40e3·38-s + 1.60e3·40-s − 1.54e3·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.169·7-s + 0.353·8-s + 0.316·10-s + 1.91·11-s − 0.0754·13-s − 0.119·14-s + 1/4·16-s − 0.317·17-s + 0.699·19-s + 0.223·20-s + 1.35·22-s + 0.782·23-s + 1/5·25-s − 0.0533·26-s − 0.0848·28-s + 1.23·29-s − 0.745·31-s + 0.176·32-s − 0.224·34-s − 0.0758·35-s − 0.0170·37-s + 0.494·38-s + 0.158·40-s − 0.143·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(90\)    =    \(2 \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{90} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 90,\ (\ :5/2),\ 1)$
$L(3)$  $\approx$  $3.22788$
$L(\frac12)$  $\approx$  $3.22788$
$L(\frac{7}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) 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 \( 1 - p^{2} T \)
3 \( 1 \)
5 \( 1 - p^{2} T \)
good7 \( 1 + 22 T + p^{5} T^{2} \)
11 \( 1 - 768 T + p^{5} T^{2} \)
13 \( 1 + 46 T + p^{5} T^{2} \)
17 \( 1 + 378 T + p^{5} T^{2} \)
19 \( 1 - 1100 T + p^{5} T^{2} \)
23 \( 1 - 1986 T + p^{5} T^{2} \)
29 \( 1 - 5610 T + p^{5} T^{2} \)
31 \( 1 + 3988 T + p^{5} T^{2} \)
37 \( 1 + 142 T + p^{5} T^{2} \)
41 \( 1 + 1542 T + p^{5} T^{2} \)
43 \( 1 + 5026 T + p^{5} T^{2} \)
47 \( 1 + 24738 T + p^{5} T^{2} \)
53 \( 1 - 14166 T + p^{5} T^{2} \)
59 \( 1 + 28380 T + p^{5} T^{2} \)
61 \( 1 - 5522 T + p^{5} T^{2} \)
67 \( 1 + 24742 T + p^{5} T^{2} \)
71 \( 1 + 42372 T + p^{5} T^{2} \)
73 \( 1 + 52126 T + p^{5} T^{2} \)
79 \( 1 + 39640 T + p^{5} T^{2} \)
83 \( 1 - 59826 T + p^{5} T^{2} \)
89 \( 1 + 57690 T + p^{5} T^{2} \)
97 \( 1 + 144382 T + p^{5} T^{2} \)
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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

−13.24666158483134573823380041282, −12.11946985556145337741053956107, −11.27754051671360354964741768362, −9.865688314231335110745992852023, −8.814289414596781130636605358233, −7.04118951844930472381286173700, −6.14710749307038265541751497190, −4.67524195817721210555723750364, −3.27151233735798890991710105439, −1.44211907248324422909812481473, 1.44211907248324422909812481473, 3.27151233735798890991710105439, 4.67524195817721210555723750364, 6.14710749307038265541751497190, 7.04118951844930472381286173700, 8.814289414596781130636605358233, 9.865688314231335110745992852023, 11.27754051671360354964741768362, 12.11946985556145337741053956107, 13.24666158483134573823380041282

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