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  − 1.51·2-s + 5.95·3-s − 5.69·4-s − 9.03·6-s + 3.35·7-s + 20.7·8-s + 8.40·9-s + 20.2·11-s − 33.8·12-s + 5.57·13-s − 5.09·14-s + 13.9·16-s + 26.5·17-s − 12.7·18-s − 25.3·19-s + 19.9·21-s − 30.7·22-s + 90.2·23-s + 123.·24-s − 8.46·26-s − 110.·27-s − 19.1·28-s − 123.·29-s + 129.·31-s − 187.·32-s + 120.·33-s − 40.2·34-s + ⋯
L(s)  = 1  − 0.536·2-s + 1.14·3-s − 0.711·4-s − 0.614·6-s + 0.181·7-s + 0.919·8-s + 0.311·9-s + 0.554·11-s − 0.814·12-s + 0.118·13-s − 0.0973·14-s + 0.218·16-s + 0.378·17-s − 0.167·18-s − 0.306·19-s + 0.207·21-s − 0.297·22-s + 0.818·23-s + 1.05·24-s − 0.0638·26-s − 0.788·27-s − 0.128·28-s − 0.789·29-s + 0.750·31-s − 1.03·32-s + 0.634·33-s − 0.203·34-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  =  $(2,\ 1175,\ (\ :3/2),\ 1)$
$L(2)$  $\approx$  $2.082544401$
$L(\frac12)$  $\approx$  $2.082544401$
$L(\frac{5}{2})$   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 \( 1 + 1.51T + 8T^{2} \)
3 \( 1 - 5.95T + 27T^{2} \)
7 \( 1 - 3.35T + 343T^{2} \)
11 \( 1 - 20.2T + 1.33e3T^{2} \)
13 \( 1 - 5.57T + 2.19e3T^{2} \)
17 \( 1 - 26.5T + 4.91e3T^{2} \)
19 \( 1 + 25.3T + 6.85e3T^{2} \)
23 \( 1 - 90.2T + 1.21e4T^{2} \)
29 \( 1 + 123.T + 2.43e4T^{2} \)
31 \( 1 - 129.T + 2.97e4T^{2} \)
37 \( 1 - 213.T + 5.06e4T^{2} \)
41 \( 1 + 124.T + 6.89e4T^{2} \)
43 \( 1 - 424.T + 7.95e4T^{2} \)
53 \( 1 + 361.T + 1.48e5T^{2} \)
59 \( 1 - 836.T + 2.05e5T^{2} \)
61 \( 1 + 194.T + 2.26e5T^{2} \)
67 \( 1 + 902.T + 3.00e5T^{2} \)
71 \( 1 - 690.T + 3.57e5T^{2} \)
73 \( 1 - 698.T + 3.89e5T^{2} \)
79 \( 1 + 449.T + 4.93e5T^{2} \)
83 \( 1 - 543.T + 5.71e5T^{2} \)
89 \( 1 - 725.T + 7.04e5T^{2} \)
97 \( 1 - 214.T + 9.12e5T^{2} \)
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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

−9.273604700508178925567952752779, −8.661535341509424576294226178845, −7.994032970991170218435366373119, −7.33681484205522815136160844077, −6.10510903025516307938248923744, −4.94438638334739554779502450213, −4.02470683484685360097751601046, −3.17247929043010025150706433091, −1.94827046596921416600186491710, −0.789840490471609575049896819847, 0.789840490471609575049896819847, 1.94827046596921416600186491710, 3.17247929043010025150706433091, 4.02470683484685360097751601046, 4.94438638334739554779502450213, 6.10510903025516307938248923744, 7.33681484205522815136160844077, 7.994032970991170218435366373119, 8.661535341509424576294226178845, 9.273604700508178925567952752779

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