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Properties

Label 2-72-72.43-c2-0-0
Degree $2$
Conductor $72$
Sign $-0.679 - 0.733i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.104 − 1.99i)2-s + (−2.08 + 2.15i)3-s + (−3.97 − 0.417i)4-s + (−5.42 + 3.12i)5-s + (4.08 + 4.39i)6-s + (−5.96 − 3.44i)7-s + (−1.24 + 7.90i)8-s + (−0.284 − 8.99i)9-s + (5.68 + 11.1i)10-s + (−5.38 + 9.32i)11-s + (9.20 − 7.70i)12-s + (15.3 − 8.88i)13-s + (−7.49 + 11.5i)14-s + (4.57 − 18.2i)15-s + (15.6 + 3.32i)16-s − 0.681·17-s + ⋯
L(s)  = 1  + (0.0522 − 0.998i)2-s + (−0.695 + 0.718i)3-s + (−0.994 − 0.104i)4-s + (−1.08 + 0.625i)5-s + (0.680 + 0.732i)6-s + (−0.851 − 0.491i)7-s + (−0.156 + 0.987i)8-s + (−0.0316 − 0.999i)9-s + (0.568 + 1.11i)10-s + (−0.489 + 0.847i)11-s + (0.766 − 0.641i)12-s + (1.18 − 0.683i)13-s + (−0.535 + 0.825i)14-s + (0.304 − 1.21i)15-s + (0.978 + 0.207i)16-s − 0.0400·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.679 - 0.733i$
Analytic conductor: \(1.96185\)
Root analytic conductor: \(1.40066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1),\ -0.679 - 0.733i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0444612 + 0.101836i\)
\(L(\frac12)\) \(\approx\) \(0.0444612 + 0.101836i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.104 + 1.99i)T \)
3 \( 1 + (2.08 - 2.15i)T \)
good5 \( 1 + (5.42 - 3.12i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (5.96 + 3.44i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (5.38 - 9.32i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-15.3 + 8.88i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 0.681T + 289T^{2} \)
19 \( 1 + 26.6T + 361T^{2} \)
23 \( 1 + (22.8 - 13.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (26.8 + 15.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-19.9 + 11.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 33.4iT - 1.36e3T^{2} \)
41 \( 1 + (-13.4 - 23.3i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (16.5 - 28.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-10.6 - 6.13i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 49.2iT - 2.80e3T^{2} \)
59 \( 1 + (19.0 + 32.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (65.7 + 37.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (27.4 + 47.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 35.6iT - 5.04e3T^{2} \)
73 \( 1 - 113.T + 5.32e3T^{2} \)
79 \( 1 + (30.8 + 17.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-58.7 + 101. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 58.3T + 7.92e3T^{2} \)
97 \( 1 + (63.7 - 110. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−15.03686636633460688331197486771, −13.36075644436065285463916879820, −12.37838163338165390072094131657, −11.26842416118601186258617034483, −10.55681094135843138379001332989, −9.668065501482805415950591856916, −7.997765690416019784588280207836, −6.20402568071269653829645241710, −4.33732126528429257938806335550, −3.40877680862397527561988827062, 0.10227426866284063852597358306, 4.07169681120644469726652554596, 5.72831784055532400912561343456, 6.65582351671324724698350445413, 8.099127090381049123349592686005, 8.819744984540086452685487752892, 10.79342889479107082367282881562, 12.15930521421365411273029572795, 12.89232364132578326547719584986, 13.85464071368264489341483603509

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