Properties

Label 2-525-1.1-c3-0-42
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.88·2-s + 3·3-s + 15.9·4-s + 14.6·6-s − 7·7-s + 38.6·8-s + 9·9-s + 54.9·11-s + 47.7·12-s + 49.7·13-s − 34.2·14-s + 61.7·16-s − 133.·17-s + 44.0·18-s + 138.·19-s − 21·21-s + 268.·22-s − 7.32·23-s + 115.·24-s + 243.·26-s + 27·27-s − 111.·28-s + 87.2·29-s − 209.·31-s − 7.32·32-s + 164.·33-s − 653.·34-s + ⋯
L(s)  = 1  + 1.72·2-s + 0.577·3-s + 1.98·4-s + 0.998·6-s − 0.377·7-s + 1.70·8-s + 0.333·9-s + 1.50·11-s + 1.14·12-s + 1.06·13-s − 0.653·14-s + 0.964·16-s − 1.90·17-s + 0.576·18-s + 1.67·19-s − 0.218·21-s + 2.60·22-s − 0.0664·23-s + 0.986·24-s + 1.83·26-s + 0.192·27-s − 0.751·28-s + 0.558·29-s − 1.21·31-s − 0.0404·32-s + 0.868·33-s − 3.29·34-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: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.208358841\)
\(L(\frac12)\) \(\approx\) \(7.208358841\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.88T + 8T^{2} \)
11 \( 1 - 54.9T + 1.33e3T^{2} \)
13 \( 1 - 49.7T + 2.19e3T^{2} \)
17 \( 1 + 133.T + 4.91e3T^{2} \)
19 \( 1 - 138.T + 6.85e3T^{2} \)
23 \( 1 + 7.32T + 1.21e4T^{2} \)
29 \( 1 - 87.2T + 2.43e4T^{2} \)
31 \( 1 + 209.T + 2.97e4T^{2} \)
37 \( 1 - 67.9T + 5.06e4T^{2} \)
41 \( 1 - 77.6T + 6.89e4T^{2} \)
43 \( 1 - 197.T + 7.95e4T^{2} \)
47 \( 1 + 4.97T + 1.03e5T^{2} \)
53 \( 1 - 53.0T + 1.48e5T^{2} \)
59 \( 1 + 683.T + 2.05e5T^{2} \)
61 \( 1 + 26.8T + 2.26e5T^{2} \)
67 \( 1 + 149.T + 3.00e5T^{2} \)
71 \( 1 - 6.15T + 3.57e5T^{2} \)
73 \( 1 + 294.T + 3.89e5T^{2} \)
79 \( 1 + 938.T + 4.93e5T^{2} \)
83 \( 1 + 784.T + 5.71e5T^{2} \)
89 \( 1 - 275.T + 7.04e5T^{2} \)
97 \( 1 + 1.16e3T + 9.12e5T^{2} \)
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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

−10.92290017569781486699172988488, −9.437142264859819901687027020668, −8.783942544354609298711670297400, −7.28573962829621897031868380717, −6.56282710405041460960802856145, −5.78114807145615486758374856344, −4.44802316259407909141918531504, −3.79449963010170565958273280890, −2.88187889879734861122739696310, −1.53067848342335046443945344614, 1.53067848342335046443945344614, 2.88187889879734861122739696310, 3.79449963010170565958273280890, 4.44802316259407909141918531504, 5.78114807145615486758374856344, 6.56282710405041460960802856145, 7.28573962829621897031868380717, 8.783942544354609298711670297400, 9.437142264859819901687027020668, 10.92290017569781486699172988488

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