Properties

Label 2-975-1.1-c1-0-34
Degree $2$
Conductor $975$
Sign $-1$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + 3-s − 2·4-s − 7-s + 9-s − 11-s − 2·12-s − 13-s + 4·16-s − 17-s − 4·19-s − 21-s − 3·23-s + 27-s + 2·28-s − 8·29-s − 4·31-s − 33-s − 2·36-s + 3·37-s − 39-s − 9·41-s − 8·43-s + 2·44-s + 10·47-s + 4·48-s − 6·49-s − 51-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s − 0.242·17-s − 0.917·19-s − 0.218·21-s − 0.625·23-s + 0.192·27-s + 0.377·28-s − 1.48·29-s − 0.718·31-s − 0.174·33-s − 1/3·36-s + 0.493·37-s − 0.160·39-s − 1.40·41-s − 1.21·43-s + 0.301·44-s + 1.45·47-s + 0.577·48-s − 6/7·49-s − 0.140·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
13 \( 1 + T \)
good2 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−9.532862510891535243606980619416, −8.790775602793191528061887163246, −8.095863304798581886113130196548, −7.22491697018544276884142731851, −6.09052611465894365795243669910, −5.09006707574467841333534810174, −4.13458107803479074584674344701, −3.31456654216549001444250807099, −1.94623851103131791750784419509, 0, 1.94623851103131791750784419509, 3.31456654216549001444250807099, 4.13458107803479074584674344701, 5.09006707574467841333534810174, 6.09052611465894365795243669910, 7.22491697018544276884142731851, 8.095863304798581886113130196548, 8.790775602793191528061887163246, 9.532862510891535243606980619416

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