Properties

Label 2-1575-1.1-c1-0-17
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 7-s − 3·8-s + 6·13-s − 14-s − 16-s + 2·17-s − 8·19-s + 8·23-s + 6·26-s + 28-s + 2·29-s + 4·31-s + 5·32-s + 2·34-s + 2·37-s − 8·38-s + 6·41-s − 4·43-s + 8·46-s + 8·47-s + 49-s − 6·52-s + 10·53-s + 3·56-s + 2·58-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s − 1.83·19-s + 1.66·23-s + 1.17·26-s + 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.883·32-s + 0.342·34-s + 0.328·37-s − 1.29·38-s + 0.937·41-s − 0.609·43-s + 1.17·46-s + 1.16·47-s + 1/7·49-s − 0.832·52-s + 1.37·53-s + 0.400·56-s + 0.262·58-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: $\chi_{1575} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.963829443\)
\(L(\frac12)\) \(\approx\) \(1.963829443\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−9.161169131924907864593244440765, −8.789607307934368613392395430343, −7.977292875613882940498903211299, −6.63585561999092413112257281527, −6.16149878863294527126110458225, −5.26003387597415288810047334583, −4.28899163544686891904773944334, −3.63296121870362536460359180450, −2.64765980094137076229438805768, −0.917037574559643692523689356640, 0.917037574559643692523689356640, 2.64765980094137076229438805768, 3.63296121870362536460359180450, 4.28899163544686891904773944334, 5.26003387597415288810047334583, 6.16149878863294527126110458225, 6.63585561999092413112257281527, 7.977292875613882940498903211299, 8.789607307934368613392395430343, 9.161169131924907864593244440765

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