Properties

Label 2-2925-1.1-c1-0-21
Degree $2$
Conductor $2925$
Sign $1$
Analytic cond. $23.3562$
Root an. cond. $4.83282$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 7-s + 11-s + 13-s + 4·16-s − 17-s − 4·19-s − 3·23-s − 2·28-s + 8·29-s − 4·31-s − 3·37-s + 9·41-s + 8·43-s − 2·44-s + 10·47-s − 6·49-s − 2·52-s − 53-s − 4·59-s − 11·61-s − 8·64-s + 4·67-s + 2·68-s + 71-s − 14·73-s + 8·76-s + ⋯
L(s)  = 1  − 4-s + 0.377·7-s + 0.301·11-s + 0.277·13-s + 16-s − 0.242·17-s − 0.917·19-s − 0.625·23-s − 0.377·28-s + 1.48·29-s − 0.718·31-s − 0.493·37-s + 1.40·41-s + 1.21·43-s − 0.301·44-s + 1.45·47-s − 6/7·49-s − 0.277·52-s − 0.137·53-s − 0.520·59-s − 1.40·61-s − 64-s + 0.488·67-s + 0.242·68-s + 0.118·71-s − 1.63·73-s + 0.917·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(23.3562\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.371039565\)
\(L(\frac12)\) \(\approx\) \(1.371039565\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good2 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.897663172163547484325799351713, −8.089771937793132184499294415351, −7.43759897448906982672581790235, −6.30669141510903739665096402460, −5.73278414244578144806247587705, −4.62157204419073793031116621923, −4.26067299548702583887435194086, −3.24149746825360535310979514555, −1.99562059501787958433191990418, −0.73375372520195490235059592257, 0.73375372520195490235059592257, 1.99562059501787958433191990418, 3.24149746825360535310979514555, 4.26067299548702583887435194086, 4.62157204419073793031116621923, 5.73278414244578144806247587705, 6.30669141510903739665096402460, 7.43759897448906982672581790235, 8.089771937793132184499294415351, 8.897663172163547484325799351713

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