Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·29-s + 2·41-s + 49-s − 2·61-s + 2·89-s − 2·101-s + 2·109-s + ⋯
L(s)  = 1  + 2·29-s + 2·41-s + 49-s − 2·61-s + 2·89-s − 2·101-s + 2·109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{3600} (3151, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3600,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $1.306934344$
$L(\frac12)$  $\approx$  $1.306934344$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )^{2} \)
97 \( 1 + T^{2} \)
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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

−8.765174895244531193974508901748, −7.964501655175975547360355392106, −7.32128235563975340193434195132, −6.43913369791821294170616881749, −5.84346598347673950005578099283, −4.84522398827095556756981450152, −4.21012682675798903946459521558, −3.15039301255661866883959383893, −2.33208052629464066896006072670, −1.03180259769072802093206375626, 1.03180259769072802093206375626, 2.33208052629464066896006072670, 3.15039301255661866883959383893, 4.21012682675798903946459521558, 4.84522398827095556756981450152, 5.84346598347673950005578099283, 6.43913369791821294170616881749, 7.32128235563975340193434195132, 7.964501655175975547360355392106, 8.765174895244531193974508901748

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