Properties

Label 2-575-1.1-c1-0-14
Degree $2$
Conductor $575$
Sign $-1$
Analytic cond. $4.59139$
Root an. cond. $2.14275$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s + 4·6-s − 7-s + 9-s − 4·12-s − 2·13-s + 2·14-s − 4·16-s + 5·17-s − 2·18-s + 8·19-s + 2·21-s + 23-s + 4·26-s + 4·27-s − 2·28-s − 5·29-s − 5·31-s + 8·32-s − 10·34-s + 2·36-s − 7·37-s − 16·38-s + 4·39-s − 7·41-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s − 0.377·7-s + 1/3·9-s − 1.15·12-s − 0.554·13-s + 0.534·14-s − 16-s + 1.21·17-s − 0.471·18-s + 1.83·19-s + 0.436·21-s + 0.208·23-s + 0.784·26-s + 0.769·27-s − 0.377·28-s − 0.928·29-s − 0.898·31-s + 1.41·32-s − 1.71·34-s + 1/3·36-s − 1.15·37-s − 2.59·38-s + 0.640·39-s − 1.09·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(4.59139\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 575,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−10.08956543988940995050722800462, −9.654091189222631032501827269352, −8.654319095271452301047967713911, −7.53022694834992702662478343447, −7.00155655462405775455941547341, −5.73625739555288291436469619815, −5.01098694924706036592409992924, −3.25732449518046581438468298305, −1.39515859044792359909937567271, 0, 1.39515859044792359909937567271, 3.25732449518046581438468298305, 5.01098694924706036592409992924, 5.73625739555288291436469619815, 7.00155655462405775455941547341, 7.53022694834992702662478343447, 8.654319095271452301047967713911, 9.654091189222631032501827269352, 10.08956543988940995050722800462

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