Properties

Label 2-972-3.2-c0-0-1
Degree $2$
Conductor $972$
Sign $1$
Analytic cond. $0.485091$
Root an. cond. $0.696484$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 13-s − 19-s + 25-s − 31-s + 2·37-s − 43-s + 3·49-s − 61-s − 67-s − 73-s − 79-s − 2·91-s − 97-s − 103-s − 109-s + ⋯
L(s)  = 1  + 2·7-s − 13-s − 19-s + 25-s − 31-s + 2·37-s − 43-s + 3·49-s − 61-s − 67-s − 73-s − 79-s − 2·91-s − 97-s − 103-s − 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(972\)    =    \(2^{2} \cdot 3^{5}\)
Sign: $1$
Analytic conductor: \(0.485091\)
Root analytic conductor: \(0.696484\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{972} (485, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 972,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.177327530\)
\(L(\frac12)\) \(\approx\) \(1.177327530\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−10.39910176385937557434601874012, −9.275682557629399676683555787935, −8.446662139628207584822994120489, −7.78364377796361895561306890859, −7.02105401856216215121476567127, −5.75752432813330098306648846518, −4.83167465577236058025265780388, −4.29236787414528275602366773914, −2.62320071892465479723209549447, −1.56538972882346577250590358999, 1.56538972882346577250590358999, 2.62320071892465479723209549447, 4.29236787414528275602366773914, 4.83167465577236058025265780388, 5.75752432813330098306648846518, 7.02105401856216215121476567127, 7.78364377796361895561306890859, 8.446662139628207584822994120489, 9.275682557629399676683555787935, 10.39910176385937557434601874012

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