Properties

Label 2-72-1.1-c7-0-7
Degree $2$
Conductor $72$
Sign $-1$
Analytic cond. $22.4917$
Root an. cond. $4.74254$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 82·5-s − 456·7-s + 2.52e3·11-s − 1.07e4·13-s + 1.11e4·17-s + 4.12e3·19-s − 8.17e4·23-s − 7.14e4·25-s − 9.97e4·29-s − 4.04e4·31-s − 3.73e4·35-s − 4.19e5·37-s − 1.41e5·41-s − 6.90e5·43-s + 6.82e5·47-s − 6.15e5·49-s − 1.81e6·53-s + 2.06e5·55-s + 9.66e5·59-s + 1.88e6·61-s − 8.83e5·65-s + 2.96e6·67-s + 2.54e6·71-s − 1.68e6·73-s − 1.15e6·77-s + 4.03e6·79-s + 5.38e6·83-s + ⋯
L(s)  = 1  + 0.293·5-s − 0.502·7-s + 0.571·11-s − 1.36·13-s + 0.550·17-s + 0.137·19-s − 1.40·23-s − 0.913·25-s − 0.759·29-s − 0.244·31-s − 0.147·35-s − 1.36·37-s − 0.320·41-s − 1.32·43-s + 0.958·47-s − 0.747·49-s − 1.67·53-s + 0.167·55-s + 0.612·59-s + 1.06·61-s − 0.399·65-s + 1.20·67-s + 0.844·71-s − 0.505·73-s − 0.287·77-s + 0.921·79-s + 1.03·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(22.4917\)
Root analytic conductor: \(4.74254\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 72,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 82 T + p^{7} T^{2} \)
7 \( 1 + 456 T + p^{7} T^{2} \)
11 \( 1 - 2524 T + p^{7} T^{2} \)
13 \( 1 + 10778 T + p^{7} T^{2} \)
17 \( 1 - 11150 T + p^{7} T^{2} \)
19 \( 1 - 4124 T + p^{7} T^{2} \)
23 \( 1 + 81704 T + p^{7} T^{2} \)
29 \( 1 + 99798 T + p^{7} T^{2} \)
31 \( 1 + 40480 T + p^{7} T^{2} \)
37 \( 1 + 419442 T + p^{7} T^{2} \)
41 \( 1 + 141402 T + p^{7} T^{2} \)
43 \( 1 + 690428 T + p^{7} T^{2} \)
47 \( 1 - 682032 T + p^{7} T^{2} \)
53 \( 1 + 1813118 T + p^{7} T^{2} \)
59 \( 1 - 966028 T + p^{7} T^{2} \)
61 \( 1 - 1887670 T + p^{7} T^{2} \)
67 \( 1 - 2965868 T + p^{7} T^{2} \)
71 \( 1 - 2548232 T + p^{7} T^{2} \)
73 \( 1 + 1680326 T + p^{7} T^{2} \)
79 \( 1 - 4038064 T + p^{7} T^{2} \)
83 \( 1 - 5385764 T + p^{7} T^{2} \)
89 \( 1 - 6473046 T + p^{7} T^{2} \)
97 \( 1 + 6065758 T + p^{7} T^{2} \)
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

−12.54750675928819810743251416885, −11.71382253108979833362580265536, −10.12680396151686957834151089507, −9.442974138710198574329240784141, −7.88817169541942228942409990252, −6.62634796092935505527386980096, −5.30011887830093967411567531193, −3.65267360208996973439063462754, −1.98006342276864053313501489003, 0, 1.98006342276864053313501489003, 3.65267360208996973439063462754, 5.30011887830093967411567531193, 6.62634796092935505527386980096, 7.88817169541942228942409990252, 9.442974138710198574329240784141, 10.12680396151686957834151089507, 11.71382253108979833362580265536, 12.54750675928819810743251416885

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