Properties

Label 2-72-1.1-c9-0-7
Degree $2$
Conductor $72$
Sign $-1$
Analytic cond. $37.0825$
Root an. cond. $6.08954$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 830·5-s + 672·7-s + 7.34e4·11-s − 7.82e4·13-s + 1.61e5·17-s − 6.53e5·19-s + 1.06e6·23-s − 1.26e6·25-s − 3.82e6·29-s − 1.57e6·31-s − 5.57e5·35-s + 1.60e7·37-s − 2.62e7·41-s − 4.44e7·43-s − 1.43e7·47-s − 3.99e7·49-s + 2.43e7·53-s − 6.09e7·55-s − 1.19e7·59-s − 1.89e8·61-s + 6.49e7·65-s − 1.06e8·67-s − 3.02e8·71-s + 8.17e7·73-s + 4.93e7·77-s + 3.15e8·79-s − 7.52e8·83-s + ⋯
L(s)  = 1  − 0.593·5-s + 0.105·7-s + 1.51·11-s − 0.759·13-s + 0.469·17-s − 1.15·19-s + 0.794·23-s − 0.647·25-s − 1.00·29-s − 0.307·31-s − 0.0628·35-s + 1.40·37-s − 1.45·41-s − 1.98·43-s − 0.428·47-s − 0.988·49-s + 0.424·53-s − 0.898·55-s − 0.128·59-s − 1.75·61-s + 0.451·65-s − 0.646·67-s − 1.41·71-s + 0.337·73-s + 0.160·77-s + 0.910·79-s − 1.74·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(37.0825\)
Root analytic conductor: \(6.08954\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 72,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 + 166 p T + p^{9} T^{2} \)
7 \( 1 - 96 p T + p^{9} T^{2} \)
11 \( 1 - 73468 T + p^{9} T^{2} \)
13 \( 1 + 78242 T + p^{9} T^{2} \)
17 \( 1 - 161726 T + p^{9} T^{2} \)
19 \( 1 + 653572 T + p^{9} T^{2} \)
23 \( 1 - 1066696 T + p^{9} T^{2} \)
29 \( 1 + 3824838 T + p^{9} T^{2} \)
31 \( 1 + 1579480 T + p^{9} T^{2} \)
37 \( 1 - 16015590 T + p^{9} T^{2} \)
41 \( 1 + 26268282 T + p^{9} T^{2} \)
43 \( 1 + 44495228 T + p^{9} T^{2} \)
47 \( 1 + 14324160 T + p^{9} T^{2} \)
53 \( 1 - 24386050 T + p^{9} T^{2} \)
59 \( 1 + 11942084 T + p^{9} T^{2} \)
61 \( 1 + 189740258 T + p^{9} T^{2} \)
67 \( 1 + 106709572 T + p^{9} T^{2} \)
71 \( 1 + 302754376 T + p^{9} T^{2} \)
73 \( 1 - 81769546 T + p^{9} T^{2} \)
79 \( 1 - 315315352 T + p^{9} T^{2} \)
83 \( 1 + 752833276 T + p^{9} T^{2} \)
89 \( 1 - 433284294 T + p^{9} T^{2} \)
97 \( 1 - 1282496642 T + p^{9} T^{2} \)
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

−12.04685437961247123460140832083, −11.31229761757030501748781120242, −9.874101893585847178327941689600, −8.753836943100851189773169888051, −7.49461424692404120027818803256, −6.32320432735280900359594019856, −4.66186524437132029792808857672, −3.47286057320199859281825363907, −1.65121520812514296804937578064, 0, 1.65121520812514296804937578064, 3.47286057320199859281825363907, 4.66186524437132029792808857672, 6.32320432735280900359594019856, 7.49461424692404120027818803256, 8.753836943100851189773169888051, 9.874101893585847178327941689600, 11.31229761757030501748781120242, 12.04685437961247123460140832083

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