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Properties

Label 2-72-72.11-c3-0-16
Degree $2$
Conductor $72$
Sign $0.828 + 0.560i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 − 0.250i)2-s + (2.93 − 4.28i)3-s + (7.87 + 1.41i)4-s + (3.52 + 6.10i)5-s + (−9.34 + 11.3i)6-s + (16.9 + 9.76i)7-s + (−21.8 − 5.95i)8-s + (−9.74 − 25.1i)9-s + (−8.39 − 18.0i)10-s + (11.8 + 6.82i)11-s + (29.1 − 29.6i)12-s + (55.8 − 32.2i)13-s + (−45.1 − 31.7i)14-s + (36.5 + 2.82i)15-s + (60.0 + 22.2i)16-s − 47.8i·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0886i)2-s + (0.565 − 0.824i)3-s + (0.984 + 0.176i)4-s + (0.315 + 0.545i)5-s + (−0.636 + 0.771i)6-s + (0.912 + 0.526i)7-s + (−0.964 − 0.263i)8-s + (−0.360 − 0.932i)9-s + (−0.265 − 0.571i)10-s + (0.324 + 0.187i)11-s + (0.701 − 0.712i)12-s + (1.19 − 0.688i)13-s + (−0.862 − 0.605i)14-s + (0.628 + 0.0485i)15-s + (0.937 + 0.347i)16-s − 0.682i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.828 + 0.560i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ 0.828 + 0.560i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.26718 - 0.388500i\)
\(L(\frac12)\) \(\approx\) \(1.26718 - 0.388500i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.81 + 0.250i)T \)
3 \( 1 + (-2.93 + 4.28i)T \)
good5 \( 1 + (-3.52 - 6.10i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-16.9 - 9.76i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-11.8 - 6.82i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-55.8 + 32.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 47.8iT - 4.91e3T^{2} \)
19 \( 1 - 14.9T + 6.85e3T^{2} \)
23 \( 1 + (25.1 + 43.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (129. - 223. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (108. - 62.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 208. iT - 5.06e4T^{2} \)
41 \( 1 + (-411. + 237. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (178. - 309. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (156. - 270. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 476.T + 1.48e5T^{2} \)
59 \( 1 + (425. - 245. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (661. + 381. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-277. - 479. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 87.1T + 3.57e5T^{2} \)
73 \( 1 + 818.T + 3.89e5T^{2} \)
79 \( 1 + (1.03e3 + 594. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-400. - 231. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (562. - 974. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.24364315090304079510645404583, −12.80106261942985090020717649698, −11.66165784761887406583944884007, −10.69173305921586925428221235588, −9.171821517780673536526145976024, −8.311721179149492559567595723724, −7.21186862256732761584301775769, −5.98659002732998312062778470233, −2.96526371793166136712906462397, −1.46958693924491494361389964844, 1.66784949881689901110724662681, 3.97532315855364596233370259629, 5.77303374498011612686724188987, 7.66073645914261858871946133513, 8.681336774229029327529988755169, 9.473266289271113016113335290018, 10.76053689719692161372483550416, 11.47727084676069518710640479295, 13.39254339745812470052544795773, 14.47254053867447075251933589892

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