Properties

Label 2-525-1.1-c1-0-15
Degree $2$
Conductor $525$
Sign $-1$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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-s + 3-s − 4-s − 6-s − 7-s + 3·8-s + 9-s − 6·11-s − 12-s − 2·13-s + 14-s − 16-s + 4·17-s − 18-s − 6·19-s − 21-s + 6·22-s + 3·24-s + 2·26-s + 27-s + 28-s − 2·29-s − 10·31-s − 5·32-s − 6·33-s − 4·34-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.37·19-s − 0.218·21-s + 1.27·22-s + 0.612·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 1.79·31-s − 0.883·32-s − 1.04·33-s − 0.685·34-s − 1/6·36-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

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−10.32936245971431996275671894173, −9.483915259822821092831866225661, −8.662795051661330682031069514757, −7.84120783606502230633349535272, −7.25875489058942760485085675987, −5.65015346036326179947249208182, −4.69294181049015431403759045817, −3.40857977083983573336135155062, −2.07938066921589939300536949382, 0, 2.07938066921589939300536949382, 3.40857977083983573336135155062, 4.69294181049015431403759045817, 5.65015346036326179947249208182, 7.25875489058942760485085675987, 7.84120783606502230633349535272, 8.662795051661330682031069514757, 9.483915259822821092831866225661, 10.32936245971431996275671894173

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