Properties

Label 2-95e2-1.1-c1-0-97
Degree $2$
Conductor $9025$
Sign $1$
Analytic cond. $72.0649$
Root an. cond. $8.48910$
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  − 2·3-s − 2·4-s + 7-s + 9-s + 3·11-s + 4·12-s − 4·13-s + 4·16-s + 3·17-s − 2·21-s + 4·27-s − 2·28-s − 6·29-s + 4·31-s − 6·33-s − 2·36-s + 2·37-s + 8·39-s + 6·41-s + 43-s − 6·44-s + 3·47-s − 8·48-s − 6·49-s − 6·51-s + 8·52-s + 12·53-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.15·12-s − 1.10·13-s + 16-s + 0.727·17-s − 0.436·21-s + 0.769·27-s − 0.377·28-s − 1.11·29-s + 0.718·31-s − 1.04·33-s − 1/3·36-s + 0.328·37-s + 1.28·39-s + 0.937·41-s + 0.152·43-s − 0.904·44-s + 0.437·47-s − 1.15·48-s − 6/7·49-s − 0.840·51-s + 1.10·52-s + 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(72.0649\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8468614901\)
\(L(\frac12)\) \(\approx\) \(0.8468614901\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.61870996574027872045561708698, −7.09492209075336668193271147178, −6.10586687521184570813995836925, −5.64391485887755959849356225387, −5.00098609153487198609192007679, −4.41624102780538194892052122814, −3.73413760442576275467138367739, −2.65038758050426083294619198007, −1.36482906238741549229048024476, −0.52679051057606925388940696432, 0.52679051057606925388940696432, 1.36482906238741549229048024476, 2.65038758050426083294619198007, 3.73413760442576275467138367739, 4.41624102780538194892052122814, 5.00098609153487198609192007679, 5.64391485887755959849356225387, 6.10586687521184570813995836925, 7.09492209075336668193271147178, 7.61870996574027872045561708698

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