Properties

Label 2-525-1.1-c3-0-50
Degree $2$
Conductor $525$
Sign $-1$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.10·2-s − 3·3-s + 8.86·4-s − 12.3·6-s − 7·7-s + 3.54·8-s + 9·9-s − 8.17·11-s − 26.5·12-s + 19.2·13-s − 28.7·14-s − 56.3·16-s + 18.8·17-s + 36.9·18-s − 76.5·19-s + 21·21-s − 33.5·22-s − 142.·23-s − 10.6·24-s + 78.9·26-s − 27·27-s − 62.0·28-s − 96.1·29-s − 270.·31-s − 259.·32-s + 24.5·33-s + 77.6·34-s + ⋯
L(s)  = 1  + 1.45·2-s − 0.577·3-s + 1.10·4-s − 0.838·6-s − 0.377·7-s + 0.156·8-s + 0.333·9-s − 0.224·11-s − 0.639·12-s + 0.410·13-s − 0.548·14-s − 0.880·16-s + 0.269·17-s + 0.483·18-s − 0.923·19-s + 0.218·21-s − 0.325·22-s − 1.29·23-s − 0.0904·24-s + 0.595·26-s − 0.192·27-s − 0.418·28-s − 0.615·29-s − 1.56·31-s − 1.43·32-s + 0.129·33-s + 0.391·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
5 \( 1 \)
7 \( 1 + 7T \)
good2 \( 1 - 4.10T + 8T^{2} \)
11 \( 1 + 8.17T + 1.33e3T^{2} \)
13 \( 1 - 19.2T + 2.19e3T^{2} \)
17 \( 1 - 18.8T + 4.91e3T^{2} \)
19 \( 1 + 76.5T + 6.85e3T^{2} \)
23 \( 1 + 142.T + 1.21e4T^{2} \)
29 \( 1 + 96.1T + 2.43e4T^{2} \)
31 \( 1 + 270.T + 2.97e4T^{2} \)
37 \( 1 - 335.T + 5.06e4T^{2} \)
41 \( 1 + 122.T + 6.89e4T^{2} \)
43 \( 1 + 492.T + 7.95e4T^{2} \)
47 \( 1 - 96.9T + 1.03e5T^{2} \)
53 \( 1 + 388.T + 1.48e5T^{2} \)
59 \( 1 - 112.T + 2.05e5T^{2} \)
61 \( 1 - 347.T + 2.26e5T^{2} \)
67 \( 1 + 101.T + 3.00e5T^{2} \)
71 \( 1 - 304.T + 3.57e5T^{2} \)
73 \( 1 - 753.T + 3.89e5T^{2} \)
79 \( 1 + 1.16e3T + 4.93e5T^{2} \)
83 \( 1 - 889.T + 5.71e5T^{2} \)
89 \( 1 + 938.T + 7.04e5T^{2} \)
97 \( 1 - 1.20e3T + 9.12e5T^{2} \)
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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

−10.27721490258472593846334379084, −9.246717001043376243220328163560, −8.001346642867866645583320358348, −6.78016870199804352023923756926, −6.05162157345321920998547532298, −5.31710995482067159098509395658, −4.24599064855875639307538210516, −3.44518004906898793119148682275, −2.03701401941646774248962406287, 0, 2.03701401941646774248962406287, 3.44518004906898793119148682275, 4.24599064855875639307538210516, 5.31710995482067159098509395658, 6.05162157345321920998547532298, 6.78016870199804352023923756926, 8.001346642867866645583320358348, 9.246717001043376243220328163560, 10.27721490258472593846334379084

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