Properties

Label 2-525-1.1-c3-0-38
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  + 1.52·2-s − 3·3-s − 5.66·4-s − 4.58·6-s − 7·7-s − 20.8·8-s + 9·9-s + 51.3·11-s + 16.9·12-s + 87.2·13-s − 10.6·14-s + 13.4·16-s − 80.6·17-s + 13.7·18-s − 29.8·19-s + 21·21-s + 78.4·22-s − 1.71·23-s + 62.6·24-s + 133.·26-s − 27·27-s + 39.6·28-s − 204.·29-s − 150.·31-s + 187.·32-s − 154.·33-s − 123.·34-s + ⋯
L(s)  = 1  + 0.540·2-s − 0.577·3-s − 0.708·4-s − 0.311·6-s − 0.377·7-s − 0.922·8-s + 0.333·9-s + 1.40·11-s + 0.408·12-s + 1.86·13-s − 0.204·14-s + 0.209·16-s − 1.15·17-s + 0.180·18-s − 0.360·19-s + 0.218·21-s + 0.760·22-s − 0.0155·23-s + 0.532·24-s + 1.00·26-s − 0.192·27-s + 0.267·28-s − 1.31·29-s − 0.870·31-s + 1.03·32-s − 0.812·33-s − 0.621·34-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 \( 1 - 1.52T + 8T^{2} \)
11 \( 1 - 51.3T + 1.33e3T^{2} \)
13 \( 1 - 87.2T + 2.19e3T^{2} \)
17 \( 1 + 80.6T + 4.91e3T^{2} \)
19 \( 1 + 29.8T + 6.85e3T^{2} \)
23 \( 1 + 1.71T + 1.21e4T^{2} \)
29 \( 1 + 204.T + 2.43e4T^{2} \)
31 \( 1 + 150.T + 2.97e4T^{2} \)
37 \( 1 + 366.T + 5.06e4T^{2} \)
41 \( 1 - 176.T + 6.89e4T^{2} \)
43 \( 1 + 394.T + 7.95e4T^{2} \)
47 \( 1 + 507.T + 1.03e5T^{2} \)
53 \( 1 - 149.T + 1.48e5T^{2} \)
59 \( 1 - 463.T + 2.05e5T^{2} \)
61 \( 1 - 380.T + 2.26e5T^{2} \)
67 \( 1 - 797.T + 3.00e5T^{2} \)
71 \( 1 + 220.T + 3.57e5T^{2} \)
73 \( 1 + 1.01e3T + 3.89e5T^{2} \)
79 \( 1 + 111.T + 4.93e5T^{2} \)
83 \( 1 + 853.T + 5.71e5T^{2} \)
89 \( 1 + 935.T + 7.04e5T^{2} \)
97 \( 1 + 783.T + 9.12e5T^{2} \)
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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

−9.996152002894902691129930557080, −8.978682722211491273031117841215, −8.573566286268104420089038905382, −6.84444533138151944876724310551, −6.22983970792024869598685961542, −5.32891851636064579769142003658, −4.05193223606088780308737484576, −3.60987410657514989769455004197, −1.50601465381175256344802350880, 0, 1.50601465381175256344802350880, 3.60987410657514989769455004197, 4.05193223606088780308737484576, 5.32891851636064579769142003658, 6.22983970792024869598685961542, 6.84444533138151944876724310551, 8.573566286268104420089038905382, 8.978682722211491273031117841215, 9.996152002894902691129930557080

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