↑

Properties

Label 2-72-72.61-c1-0-9
Degree $2$
Conductor $72$
Sign $0.146 + 0.989i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.820 − 1.15i)2-s + (−1.69 − 0.378i)3-s + (−0.654 − 1.88i)4-s + (1.97 − 1.14i)5-s + (−1.82 + 1.63i)6-s + (−0.907 + 1.57i)7-s + (−2.71 − 0.795i)8-s + (2.71 + 1.27i)9-s + (0.306 − 3.21i)10-s + (4.24 + 2.44i)11-s + (0.391 + 3.44i)12-s + (−4.00 + 2.31i)13-s + (1.06 + 2.33i)14-s + (−3.77 + 1.18i)15-s + (−3.14 + 2.47i)16-s + 1.92·17-s + ⋯
L(s)  = 1  + (0.579 − 0.814i)2-s + (−0.975 − 0.218i)3-s + (−0.327 − 0.944i)4-s + (0.883 − 0.510i)5-s + (−0.743 + 0.668i)6-s + (−0.343 + 0.594i)7-s + (−0.959 − 0.281i)8-s + (0.904 + 0.426i)9-s + (0.0968 − 1.01i)10-s + (1.27 + 0.738i)11-s + (0.113 + 0.993i)12-s + (−1.11 + 0.641i)13-s + (0.285 + 0.624i)14-s + (−0.973 + 0.304i)15-s + (−0.785 + 0.618i)16-s + 0.467·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.146 + 0.989i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.146 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.748387 - 0.645552i\)
\(L(\frac12)\) \(\approx\) \(0.748387 - 0.645552i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.820 + 1.15i)T \)
3 \( 1 + (1.69 + 0.378i)T \)
good5 \( 1 + (-1.97 + 1.14i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.907 - 1.57i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.24 - 2.44i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.00 - 2.31i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
19 \( 1 + 2.12iT - 19T^{2} \)
23 \( 1 + (1.15 + 2.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.16 + 1.82i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.65 + 4.60i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.98iT - 37T^{2} \)
41 \( 1 + (2.36 + 4.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.20 - 1.27i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.02 - 3.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.95iT - 53T^{2} \)
59 \( 1 + (3.05 - 1.76i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.71 + 0.991i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.72 + 4.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + (4.97 - 8.61i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.12 + 1.80i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.49T + 89T^{2} \)
97 \( 1 + (-6.99 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−14.15363317861161169866121288324, −12.95129972740286132079918338835, −12.22869471551045459314014241281, −11.46167129406626415193024478684, −9.852559615536150874796920553846, −9.377854038708136727060839329173, −6.75133116906701408665026130120, −5.64880768670881203338771401433, −4.50058612110268657776741812499, −1.89339069970918461618636920027, 3.71141571055986324987796352922, 5.41689759604860908302065866629, 6.34277869647692455373209935863, 7.35967194995872073588417269346, 9.351844746094888919264930361934, 10.42154244715590256735227066772, 11.81487156421440477638811344470, 12.81542388660747914613891798513, 14.02366912812648035145857858329, 14.73608679053032739279642951057

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