Properties

Label 2-1575-1.1-c1-0-43
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·2-s + 0.618·4-s − 7-s − 2.23·8-s − 4.23·11-s + 3.23·13-s − 1.61·14-s − 4.85·16-s − 6.47·17-s + 4.47·19-s − 6.85·22-s + 1.76·23-s + 5.23·26-s − 0.618·28-s − 5·29-s − 9.70·31-s − 3.38·32-s − 10.4·34-s − 3·37-s + 7.23·38-s − 9.23·41-s − 6.23·43-s − 2.61·44-s + 2.85·46-s − 2·47-s + 49-s + 2.00·52-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s − 0.377·7-s − 0.790·8-s − 1.27·11-s + 0.897·13-s − 0.432·14-s − 1.21·16-s − 1.56·17-s + 1.02·19-s − 1.46·22-s + 0.367·23-s + 1.02·26-s − 0.116·28-s − 0.928·29-s − 1.74·31-s − 0.597·32-s − 1.79·34-s − 0.493·37-s + 1.17·38-s − 1.44·41-s − 0.950·43-s − 0.394·44-s + 0.420·46-s − 0.291·47-s + 0.142·49-s + 0.277·52-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: $\chi_{1575} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 - 1.61T + 2T^{2} \)
11 \( 1 + 4.23T + 11T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
17 \( 1 + 6.47T + 17T^{2} \)
19 \( 1 - 4.47T + 19T^{2} \)
23 \( 1 - 1.76T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 9.70T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + 9.23T + 41T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + 0.472T + 53T^{2} \)
59 \( 1 - 1.70T + 59T^{2} \)
61 \( 1 - 3.70T + 61T^{2} \)
67 \( 1 + 0.236T + 67T^{2} \)
71 \( 1 - 4.70T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 5.70T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + 0.763T + 97T^{2} \)
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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

−9.016674763939395922533721064260, −8.283868080415678386542766332755, −7.15581737363023139572388353792, −6.44572772153427345416446092618, −5.42538953373525037613747107251, −5.04280082945156508818286961776, −3.83932341101615808283304805012, −3.22755375148447618726290811510, −2.09151690798107021125714979894, 0, 2.09151690798107021125714979894, 3.22755375148447618726290811510, 3.83932341101615808283304805012, 5.04280082945156508818286961776, 5.42538953373525037613747107251, 6.44572772153427345416446092618, 7.15581737363023139572388353792, 8.283868080415678386542766332755, 9.016674763939395922533721064260

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