Properties

Label 2-1575-1.1-c1-0-28
Degree $2$
Conductor $1575$
Sign $-1$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 7-s + 3·11-s + 13-s + 2·14-s − 4·16-s − 7·17-s − 6·22-s − 6·23-s − 2·26-s − 2·28-s + 5·29-s + 2·31-s + 8·32-s + 14·34-s + 2·37-s − 2·41-s − 4·43-s + 6·44-s + 12·46-s + 3·47-s + 49-s + 2·52-s − 6·53-s − 10·58-s − 10·59-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.377·7-s + 0.904·11-s + 0.277·13-s + 0.534·14-s − 16-s − 1.69·17-s − 1.27·22-s − 1.25·23-s − 0.392·26-s − 0.377·28-s + 0.928·29-s + 0.359·31-s + 1.41·32-s + 2.40·34-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.904·44-s + 1.76·46-s + 0.437·47-s + 1/7·49-s + 0.277·52-s − 0.824·53-s − 1.31·58-s − 1.30·59-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: yes
Arithmetic: yes
Character: Trivial
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 \( 1 + p T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−9.047392015771699165995183825449, −8.428704975511724884581109770971, −7.67053504463822546742181996665, −6.59661422823789620490429923129, −6.33345798756843332553352393681, −4.74217082052326318941224012916, −3.90430059874868075785507099311, −2.48112200668525509199201292270, −1.41698761693361156521107028690, 0, 1.41698761693361156521107028690, 2.48112200668525509199201292270, 3.90430059874868075785507099311, 4.74217082052326318941224012916, 6.33345798756843332553352393681, 6.59661422823789620490429923129, 7.67053504463822546742181996665, 8.428704975511724884581109770971, 9.047392015771699165995183825449

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