Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.60·2-s − 1.72·3-s − 5.41·4-s − 2.78·6-s + 11.3·7-s − 21.5·8-s − 24.0·9-s − 40.6·11-s + 9.35·12-s + 12.3·13-s + 18.2·14-s + 8.55·16-s − 59.6·17-s − 38.6·18-s + 26.4·19-s − 19.5·21-s − 65.4·22-s − 107.·23-s + 37.2·24-s + 19.9·26-s + 88.1·27-s − 61.2·28-s + 173.·29-s − 332.·31-s + 186.·32-s + 70.2·33-s − 96.0·34-s + ⋯
L(s)  = 1  + 0.568·2-s − 0.332·3-s − 0.676·4-s − 0.189·6-s + 0.611·7-s − 0.953·8-s − 0.889·9-s − 1.11·11-s + 0.224·12-s + 0.264·13-s + 0.347·14-s + 0.133·16-s − 0.851·17-s − 0.506·18-s + 0.318·19-s − 0.203·21-s − 0.633·22-s − 0.970·23-s + 0.317·24-s + 0.150·26-s + 0.628·27-s − 0.413·28-s + 1.11·29-s − 1.92·31-s + 1.02·32-s + 0.370·33-s − 0.484·34-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  =  $(2,\ 1175,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $1.077468228$
$L(\frac12)$  $\approx$  $1.077468228$
$L(\frac{5}{2})$   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 \( 1 - 1.60T + 8T^{2} \)
3 \( 1 + 1.72T + 27T^{2} \)
7 \( 1 - 11.3T + 343T^{2} \)
11 \( 1 + 40.6T + 1.33e3T^{2} \)
13 \( 1 - 12.3T + 2.19e3T^{2} \)
17 \( 1 + 59.6T + 4.91e3T^{2} \)
19 \( 1 - 26.4T + 6.85e3T^{2} \)
23 \( 1 + 107.T + 1.21e4T^{2} \)
29 \( 1 - 173.T + 2.43e4T^{2} \)
31 \( 1 + 332.T + 2.97e4T^{2} \)
37 \( 1 - 172.T + 5.06e4T^{2} \)
41 \( 1 + 178.T + 6.89e4T^{2} \)
43 \( 1 + 63.2T + 7.95e4T^{2} \)
53 \( 1 - 402.T + 1.48e5T^{2} \)
59 \( 1 - 305.T + 2.05e5T^{2} \)
61 \( 1 + 86.2T + 2.26e5T^{2} \)
67 \( 1 - 681.T + 3.00e5T^{2} \)
71 \( 1 - 726.T + 3.57e5T^{2} \)
73 \( 1 - 79.8T + 3.89e5T^{2} \)
79 \( 1 - 279.T + 4.93e5T^{2} \)
83 \( 1 + 556.T + 5.71e5T^{2} \)
89 \( 1 + 342.T + 7.04e5T^{2} \)
97 \( 1 - 1.63e3T + 9.12e5T^{2} \)
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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

−9.336855761062767642160437157799, −8.444212644849988570216417880677, −8.008367927951789945534216600230, −6.70202954299820221412985015474, −5.67016518152890794734071159370, −5.21967868114184782331678362622, −4.34691846346306906734917614577, −3.29455529101673026242604248372, −2.20621294529197593402827895188, −0.47964979178292421347621110341, 0.47964979178292421347621110341, 2.20621294529197593402827895188, 3.29455529101673026242604248372, 4.34691846346306906734917614577, 5.21967868114184782331678362622, 5.67016518152890794734071159370, 6.70202954299820221412985015474, 8.008367927951789945534216600230, 8.444212644849988570216417880677, 9.336855761062767642160437157799

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