↑

Properties

Label 2-72-72.29-c2-0-13
Degree $2$
Conductor $72$
Sign $0.557 + 0.829i$
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  + (−1.95 + 0.423i)2-s + (2.84 − 0.958i)3-s + (3.64 − 1.65i)4-s + (−3.64 − 6.31i)5-s + (−5.15 + 3.07i)6-s + (−0.487 + 0.843i)7-s + (−6.41 + 4.77i)8-s + (7.16 − 5.44i)9-s + (9.80 + 10.8i)10-s + (7.19 − 12.4i)11-s + (8.76 − 8.19i)12-s + (5.50 − 3.17i)13-s + (0.595 − 1.85i)14-s + (−16.4 − 14.4i)15-s + (10.5 − 12.0i)16-s + 27.9i·17-s + ⋯
L(s)  = 1  + (−0.977 + 0.211i)2-s + (0.947 − 0.319i)3-s + (0.910 − 0.413i)4-s + (−0.729 − 1.26i)5-s + (−0.858 + 0.512i)6-s + (−0.0695 + 0.120i)7-s + (−0.802 + 0.596i)8-s + (0.795 − 0.605i)9-s + (0.980 + 1.08i)10-s + (0.654 − 1.13i)11-s + (0.730 − 0.682i)12-s + (0.423 − 0.244i)13-s + (0.0425 − 0.132i)14-s + (−1.09 − 0.963i)15-s + (0.657 − 0.753i)16-s + 1.64i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.557 + 0.829i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ 0.557 + 0.829i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.885497 - 0.471760i\)
\(L(\frac12)\) \(\approx\) \(0.885497 - 0.471760i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.95 - 0.423i)T \)
3 \( 1 + (-2.84 + 0.958i)T \)
good5 \( 1 + (3.64 + 6.31i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (0.487 - 0.843i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-7.19 + 12.4i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.50 + 3.17i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 27.9iT - 289T^{2} \)
19 \( 1 - 9.44iT - 361T^{2} \)
23 \( 1 + (-5.15 + 2.97i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (24.9 - 43.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (11.0 + 19.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 20.4iT - 1.36e3T^{2} \)
41 \( 1 + (-16.9 + 9.77i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (19.6 + 11.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-71.3 - 41.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 53.3T + 2.80e3T^{2} \)
59 \( 1 + (29.5 + 51.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (9.09 + 5.24i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (44.8 - 25.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 11.6iT - 5.04e3T^{2} \)
73 \( 1 + 39.1T + 5.32e3T^{2} \)
79 \( 1 + (-49.7 + 86.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-3.61 + 6.26i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 97.4iT - 7.92e3T^{2} \)
97 \( 1 + (2.40 - 4.16i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−14.47177681771124460092144596665, −12.99890205867714017459766459433, −12.08570425228110066076656714331, −10.70743572210355917785220568435, −9.035848836264037928199379515596, −8.615293376963406235891763959338, −7.67426266719314574730004892588, −6.01453091065858794804412987522, −3.69546166399138888625333440339, −1.26636353998798223994084819289, 2.52014008146149170582984371258, 3.89296101661790531851245511554, 7.01180819406649925603205556402, 7.46364193968419364089619596500, 9.012292446482989583030153188938, 9.903922774638433222963829063050, 11.02525920422521368821061797242, 11.98006641253804755969856198797, 13.67850138752465944766717707572, 14.94662600806705966213551035718

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