Properties

Label 2-72-8.5-c17-0-37
Degree $2$
Conductor $72$
Sign $0.268 + 0.963i$
Analytic cond. $131.919$
Root an. cond. $11.4856$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−328. − 151. i)2-s + (8.49e4 + 9.98e4i)4-s − 6.63e5i·5-s − 1.66e7·7-s + (−1.27e7 − 4.57e7i)8-s + (−1.00e8 + 2.18e8i)10-s − 1.05e9i·11-s − 9.19e7i·13-s + (5.47e9 + 2.53e9i)14-s + (−2.75e9 + 1.69e10i)16-s + 1.98e10·17-s + 8.44e10i·19-s + (6.62e10 − 5.63e10i)20-s + (−1.60e11 + 3.46e11i)22-s + 2.72e10·23-s + ⋯
L(s)  = 1  + (−0.907 − 0.419i)2-s + (0.648 + 0.761i)4-s − 0.760i·5-s − 1.09·7-s + (−0.268 − 0.963i)8-s + (−0.318 + 0.690i)10-s − 1.48i·11-s − 0.0312i·13-s + (0.992 + 0.458i)14-s + (−0.160 + 0.987i)16-s + 0.689·17-s + 1.14i·19-s + (0.578 − 0.492i)20-s + (−0.622 + 1.34i)22-s + 0.0724·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(131.919\)
Root analytic conductor: \(11.4856\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :17/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.072636285\)
\(L(\frac12)\) \(\approx\) \(1.072636285\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (328. + 151. i)T \)
3 \( 1 \)
good5 \( 1 + 6.63e5iT - 7.62e11T^{2} \)
7 \( 1 + 1.66e7T + 2.32e14T^{2} \)
11 \( 1 + 1.05e9iT - 5.05e17T^{2} \)
13 \( 1 + 9.19e7iT - 8.65e18T^{2} \)
17 \( 1 - 1.98e10T + 8.27e20T^{2} \)
19 \( 1 - 8.44e10iT - 5.48e21T^{2} \)
23 \( 1 - 2.72e10T + 1.41e23T^{2} \)
29 \( 1 - 3.75e12iT - 7.25e24T^{2} \)
31 \( 1 - 5.36e12T + 2.25e25T^{2} \)
37 \( 1 + 1.92e13iT - 4.56e26T^{2} \)
41 \( 1 + 5.42e13T + 2.61e27T^{2} \)
43 \( 1 + 3.96e13iT - 5.87e27T^{2} \)
47 \( 1 + 4.48e13T + 2.66e28T^{2} \)
53 \( 1 - 7.81e14iT - 2.05e29T^{2} \)
59 \( 1 + 1.07e15iT - 1.27e30T^{2} \)
61 \( 1 - 1.92e15iT - 2.24e30T^{2} \)
67 \( 1 - 3.68e15iT - 1.10e31T^{2} \)
71 \( 1 - 1.02e16T + 2.96e31T^{2} \)
73 \( 1 - 1.25e16T + 4.74e31T^{2} \)
79 \( 1 - 3.24e15T + 1.81e32T^{2} \)
83 \( 1 - 4.28e15iT - 4.21e32T^{2} \)
89 \( 1 + 6.95e16T + 1.37e33T^{2} \)
97 \( 1 + 7.74e16T + 5.95e33T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.86357091226992181541230791756, −9.894814822362549640272674790084, −8.884333090671311160435163268572, −8.126835690667495910378787153761, −6.70634489999745420424748992570, −5.59738729351959112431467900658, −3.71149223809592078337043378994, −2.92742997085074562350403841464, −1.30127356289673598751760892376, −0.51741244580356292013834310932, 0.62019419958868897671522580938, 2.12214635929115119265519741697, 3.10947556132804643813177033284, 4.89282231447169857046862314026, 6.47721772736427802902051862910, 6.91424951020629161301673330772, 8.118687203732737196440092147210, 9.681618974438373260659310852560, 9.936477130031758727312714158253, 11.24342185085249883887522798690

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