Properties

Label 2-1575-1.1-c1-0-10
Degree $2$
Conductor $1575$
Sign $1$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.68·2-s + 5.19·4-s − 7-s − 8.56·8-s + 3.19·11-s + 6.38·13-s + 2.68·14-s + 12.5·16-s − 5.36·17-s − 2.38·19-s − 8.57·22-s + 3.19·23-s − 17.1·26-s − 5.19·28-s + 2.16·29-s + 6·31-s − 16.6·32-s + 14.3·34-s + 3·37-s + 6.39·38-s − 10.7·41-s − 9.38·43-s + 16.6·44-s − 8.57·46-s + 11.7·47-s + 49-s + 33.1·52-s + ⋯
L(s)  = 1  − 1.89·2-s + 2.59·4-s − 0.377·7-s − 3.02·8-s + 0.964·11-s + 1.77·13-s + 0.716·14-s + 3.14·16-s − 1.30·17-s − 0.547·19-s − 1.82·22-s + 0.666·23-s − 3.35·26-s − 0.981·28-s + 0.402·29-s + 1.07·31-s − 2.93·32-s + 2.46·34-s + 0.493·37-s + 1.03·38-s − 1.67·41-s − 1.43·43-s + 2.50·44-s − 1.26·46-s + 1.71·47-s + 0.142·49-s + 4.59·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7330294630\)
\(L(\frac12)\) \(\approx\) \(0.7330294630\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 + 2.68T + 2T^{2} \)
11 \( 1 - 3.19T + 11T^{2} \)
13 \( 1 - 6.38T + 13T^{2} \)
17 \( 1 + 5.36T + 17T^{2} \)
19 \( 1 + 2.38T + 19T^{2} \)
23 \( 1 - 3.19T + 23T^{2} \)
29 \( 1 - 2.16T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 9.38T + 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 9.38T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 - 4.38T + 73T^{2} \)
79 \( 1 + 5.38T + 79T^{2} \)
83 \( 1 - 4.33T + 83T^{2} \)
89 \( 1 + 1.03T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−9.140791689210818874293317318519, −8.719817428349082423391649808555, −8.282645142947836076338349666784, −6.94614687982193229259735236660, −6.64410108836379703747592469041, −5.88074145088887858900943359860, −4.15305800816045799105803153411, −3.03754061987304116655603212095, −1.84243654883089816851595569051, −0.820980777020587896750483728234, 0.820980777020587896750483728234, 1.84243654883089816851595569051, 3.03754061987304116655603212095, 4.15305800816045799105803153411, 5.88074145088887858900943359860, 6.64410108836379703747592469041, 6.94614687982193229259735236660, 8.282645142947836076338349666784, 8.719817428349082423391649808555, 9.140791689210818874293317318519

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