↑

Properties

Label 2-72-9.5-c2-0-2
Degree $2$
Conductor $72$
Sign $0.972 - 0.231i$
Analytic cond. $1.96185$
Root an. cond. $1.40066$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.91 + 0.690i)3-s + (1.80 + 1.04i)5-s + (−0.781 − 1.35i)7-s + (8.04 + 4.03i)9-s + (−10.8 + 6.25i)11-s + (11.0 − 19.1i)13-s + (4.56 + 4.29i)15-s + 12.6i·17-s − 21.7·19-s + (−1.34 − 4.49i)21-s + (−28.7 − 16.6i)23-s + (−10.3 − 17.8i)25-s + (20.7 + 17.3i)27-s + (−25.7 + 14.8i)29-s + (6.91 − 11.9i)31-s + ⋯
L(s)  = 1  + (0.973 + 0.230i)3-s + (0.361 + 0.208i)5-s + (−0.111 − 0.193i)7-s + (0.894 + 0.447i)9-s + (−0.984 + 0.568i)11-s + (0.849 − 1.47i)13-s + (0.304 + 0.286i)15-s + 0.747i·17-s − 1.14·19-s + (−0.0641 − 0.213i)21-s + (−1.25 − 0.722i)23-s + (−0.412 − 0.714i)25-s + (0.767 + 0.641i)27-s + (−0.888 + 0.513i)29-s + (0.223 − 0.386i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.972 - 0.231i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ 0.972 - 0.231i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.62330 + 0.190618i\)
\(L(\frac12)\) \(\approx\) \(1.62330 + 0.190618i\)
\(L(2)\) 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 \( 1 + (-2.91 - 0.690i)T \)
good5 \( 1 + (-1.80 - 1.04i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (0.781 + 1.35i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.8 - 6.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-11.0 + 19.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 12.6iT - 289T^{2} \)
19 \( 1 + 21.7T + 361T^{2} \)
23 \( 1 + (28.7 + 16.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (25.7 - 14.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.91 + 11.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 8.26T + 1.36e3T^{2} \)
41 \( 1 + (-43.8 - 25.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-35.5 - 61.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-57.2 + 33.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 6.04iT - 2.80e3T^{2} \)
59 \( 1 + (8.01 + 4.62i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-51.9 - 89.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (19.8 - 34.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 18.3iT - 5.04e3T^{2} \)
73 \( 1 + 68.5T + 5.32e3T^{2} \)
79 \( 1 + (-13.3 - 23.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-21.0 + 12.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 111. iT - 7.92e3T^{2} \)
97 \( 1 + (2.51 + 4.35i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−14.53561261970198613038060378170, −13.26842927168310108673702049346, −12.72573907536665469427860141537, −10.58587026343995250103054557201, −10.13217336690628522027131874135, −8.546951477481543028090199069956, −7.69339513761679546612270123049, −5.95438940991925571118336963073, −4.09402859165997205948647082415, −2.43714225100810203781993260795, 2.14703158511098216018464154904, 3.97257570417717186939617227475, 5.93697022809645606519218489280, 7.44570892964089504056381435444, 8.703340073771260617060150269686, 9.519332166300355612414704124818, 10.97440958989360519995612025727, 12.39039614131393729300059939207, 13.59708903976078257835287497443, 13.98131383759040998311824996493

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