Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.193·2-s − 3-s − 1.96·4-s − 0.193·6-s − 7-s − 0.768·8-s + 9-s + 2·11-s + 1.96·12-s + 1.35·13-s − 0.193·14-s + 3.77·16-s − 3.35·17-s + 0.193·18-s + 5.35·19-s + 21-s + 0.387·22-s + 4.96·23-s + 0.768·24-s + 0.261·26-s − 27-s + 1.96·28-s + 7.92·29-s + 4.57·31-s + 2.26·32-s − 2·33-s − 0.649·34-s + ⋯
L(s)  = 1  + 0.137·2-s − 0.577·3-s − 0.981·4-s − 0.0791·6-s − 0.377·7-s − 0.271·8-s + 0.333·9-s + 0.603·11-s + 0.566·12-s + 0.374·13-s − 0.0518·14-s + 0.943·16-s − 0.812·17-s + 0.0457·18-s + 1.22·19-s + 0.218·21-s + 0.0826·22-s + 1.03·23-s + 0.156·24-s + 0.0513·26-s − 0.192·27-s + 0.370·28-s + 1.47·29-s + 0.821·31-s + 0.401·32-s − 0.348·33-s − 0.111·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{525} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 525,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.997143\)
\(L(\frac12)\)  \(\approx\)  \(0.997143\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 - 0.193T + 2T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 1.35T + 13T^{2} \)
17 \( 1 + 3.35T + 17T^{2} \)
19 \( 1 - 5.35T + 19T^{2} \)
23 \( 1 - 4.96T + 23T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 - 4.57T + 31T^{2} \)
37 \( 1 + 0.775T + 37T^{2} \)
41 \( 1 - 3.73T + 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 - 9.92T + 47T^{2} \)
53 \( 1 - 8.57T + 53T^{2} \)
59 \( 1 + 8.62T + 59T^{2} \)
61 \( 1 + 8.70T + 61T^{2} \)
67 \( 1 - 9.92T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 9.35T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 3.22T + 83T^{2} \)
89 \( 1 - 1.03T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−10.83744424866424693015182860320, −9.914525071408972659806251594373, −9.156825295931349904739131572860, −8.362766972126603023432502689627, −7.06666112289600058325462250884, −6.18421278721861225303735663511, −5.14085153600714544753454493451, −4.30027321369541196988573082776, −3.14182764210480382492854285569, −0.956151394633825783233879657697, 0.956151394633825783233879657697, 3.14182764210480382492854285569, 4.30027321369541196988573082776, 5.14085153600714544753454493451, 6.18421278721861225303735663511, 7.06666112289600058325462250884, 8.362766972126603023432502689627, 9.156825295931349904739131572860, 9.914525071408972659806251594373, 10.83744424866424693015182860320

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