Properties

Degree 2
Conductor $ 5^{2} \cdot 47 $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 4.90·2-s + 0.777·3-s + 16.1·4-s + 3.81·6-s + 30.3·7-s + 39.7·8-s − 26.3·9-s + 22.4·11-s + 12.5·12-s + 62.0·13-s + 148.·14-s + 66.4·16-s + 72.1·17-s − 129.·18-s − 25.0·19-s + 23.5·21-s + 110.·22-s − 103.·23-s + 30.9·24-s + 304.·26-s − 41.5·27-s + 488.·28-s − 234.·29-s + 198.·31-s + 8.17·32-s + 17.4·33-s + 354.·34-s + ⋯
L(s)  = 1  + 1.73·2-s + 0.149·3-s + 2.01·4-s + 0.259·6-s + 1.63·7-s + 1.75·8-s − 0.977·9-s + 0.614·11-s + 0.301·12-s + 1.32·13-s + 2.84·14-s + 1.03·16-s + 1.02·17-s − 1.69·18-s − 0.302·19-s + 0.245·21-s + 1.06·22-s − 0.935·23-s + 0.263·24-s + 2.29·26-s − 0.296·27-s + 3.29·28-s − 1.50·29-s + 1.15·31-s + 0.0451·32-s + 0.0920·33-s + 1.78·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1175\)    =    \(5^{2} \cdot 47\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{1175} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1175,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $8.455390720$
$L(\frac12)$  $\approx$  $8.455390720$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;47\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{5,\;47\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 \)
47 \( 1 + 47T \)
good2 \( 1 - 4.90T + 8T^{2} \)
3 \( 1 - 0.777T + 27T^{2} \)
7 \( 1 - 30.3T + 343T^{2} \)
11 \( 1 - 22.4T + 1.33e3T^{2} \)
13 \( 1 - 62.0T + 2.19e3T^{2} \)
17 \( 1 - 72.1T + 4.91e3T^{2} \)
19 \( 1 + 25.0T + 6.85e3T^{2} \)
23 \( 1 + 103.T + 1.21e4T^{2} \)
29 \( 1 + 234.T + 2.43e4T^{2} \)
31 \( 1 - 198.T + 2.97e4T^{2} \)
37 \( 1 - 203.T + 5.06e4T^{2} \)
41 \( 1 - 210.T + 6.89e4T^{2} \)
43 \( 1 + 111.T + 7.95e4T^{2} \)
53 \( 1 + 499.T + 1.48e5T^{2} \)
59 \( 1 + 562.T + 2.05e5T^{2} \)
61 \( 1 - 548.T + 2.26e5T^{2} \)
67 \( 1 - 760.T + 3.00e5T^{2} \)
71 \( 1 + 668.T + 3.57e5T^{2} \)
73 \( 1 - 1.14e3T + 3.89e5T^{2} \)
79 \( 1 - 975.T + 4.93e5T^{2} \)
83 \( 1 + 698.T + 5.71e5T^{2} \)
89 \( 1 - 451.T + 7.04e5T^{2} \)
97 \( 1 - 390.T + 9.12e5T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.335544303255047228497124392522, −8.215761791127584658506673134632, −7.81489064801472464435350272364, −6.43829939032793818162982488721, −5.81889215223919628430680848493, −5.11281542767261898123130230537, −4.15357313851768148657754762343, −3.49711640114080504664406331807, −2.30536584016938211582356448315, −1.32077201698659161111203176339, 1.32077201698659161111203176339, 2.30536584016938211582356448315, 3.49711640114080504664406331807, 4.15357313851768148657754762343, 5.11281542767261898123130230537, 5.81889215223919628430680848493, 6.43829939032793818162982488721, 7.81489064801472464435350272364, 8.215761791127584658506673134632, 9.335544303255047228497124392522

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