Properties

Label 2-1575-7.6-c0-0-3
Degree $2$
Conductor $1575$
Sign $1$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·2-s + 1.99·4-s + 7-s − 1.73·8-s + 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99·22-s + 1.73·23-s + 1.99·28-s − 1.73·29-s − 37-s − 43-s + 3.46·44-s − 2.99·46-s + 49-s − 1.73·56-s + 2.99·58-s − 1.00·64-s + 67-s − 1.73·71-s + 1.73·74-s + 1.73·77-s − 79-s + 1.73·86-s − 2.99·88-s + 3.46·92-s + ⋯
L(s)  = 1  − 1.73·2-s + 1.99·4-s + 7-s − 1.73·8-s + 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99·22-s + 1.73·23-s + 1.99·28-s − 1.73·29-s − 37-s − 43-s + 3.46·44-s − 2.99·46-s + 49-s − 1.73·56-s + 2.99·58-s − 1.00·64-s + 67-s − 1.73·71-s + 1.73·74-s + 1.73·77-s − 79-s + 1.73·86-s − 2.99·88-s + 3.46·92-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1126, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6300909354\)
\(L(\frac12)\) \(\approx\) \(0.6300909354\)
\(L(1)\) 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 \( 1 + 1.73T + T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.73T + T^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.349615901642857061912383115461, −8.902902054120991952451699376794, −8.301686865379305276477974453446, −7.24880537699402714312724214322, −6.93558457106526796701733672629, −5.80869347817437297985492816577, −4.63449302403676709744177095760, −3.41769352103128106380981335470, −1.93960480328905945514688461578, −1.20585865558191633564296078501, 1.20585865558191633564296078501, 1.93960480328905945514688461578, 3.41769352103128106380981335470, 4.63449302403676709744177095760, 5.80869347817437297985492816577, 6.93558457106526796701733672629, 7.24880537699402714312724214322, 8.301686865379305276477974453446, 8.902902054120991952451699376794, 9.349615901642857061912383115461

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