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Properties

Label 2-72-24.11-c21-0-1
Degree $2$
Conductor $72$
Sign $-0.999 - 0.0147i$
Analytic cond. $201.223$
Root an. cond. $14.1853$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07e3 − 969. i)2-s + (2.16e5 + 2.08e6i)4-s + 2.58e6·5-s − 9.12e7i·7-s + (1.78e9 − 2.45e9i)8-s + (−2.77e9 − 2.50e9i)10-s + 1.18e10i·11-s + 5.65e11i·13-s + (−8.84e10 + 9.81e10i)14-s + (−4.30e12 + 9.03e11i)16-s + 7.94e12i·17-s + 5.23e13·19-s + (5.58e11 + 5.38e12i)20-s + (1.15e13 − 1.27e13i)22-s − 1.93e14·23-s + ⋯
L(s)  = 1  + (−0.742 − 0.669i)2-s + (0.103 + 0.994i)4-s + 0.118·5-s − 0.122i·7-s + (0.589 − 0.807i)8-s + (−0.0877 − 0.0791i)10-s + 0.137i·11-s + 1.13i·13-s + (−0.0817 + 0.0906i)14-s + (−0.978 + 0.205i)16-s + 0.956i·17-s + 1.95·19-s + (0.0122 + 0.117i)20-s + (0.0923 − 0.102i)22-s − 0.975·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0147i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.999 - 0.0147i$
Analytic conductor: \(201.223\)
Root analytic conductor: \(14.1853\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :21/2),\ -0.999 - 0.0147i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.03376332464\)
\(L(\frac12)\) \(\approx\) \(0.03376332464\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07e3 + 969. i)T \)
3 \( 1 \)
good5 \( 1 - 2.58e6T + 4.76e14T^{2} \)
7 \( 1 + 9.12e7iT - 5.58e17T^{2} \)
11 \( 1 - 1.18e10iT - 7.40e21T^{2} \)
13 \( 1 - 5.65e11iT - 2.47e23T^{2} \)
17 \( 1 - 7.94e12iT - 6.90e25T^{2} \)
19 \( 1 - 5.23e13T + 7.14e26T^{2} \)
23 \( 1 + 1.93e14T + 3.94e28T^{2} \)
29 \( 1 + 3.66e15T + 5.13e30T^{2} \)
31 \( 1 + 5.74e15iT - 2.08e31T^{2} \)
37 \( 1 - 3.31e16iT - 8.55e32T^{2} \)
41 \( 1 - 9.75e16iT - 7.38e33T^{2} \)
43 \( 1 + 1.41e17T + 2.00e34T^{2} \)
47 \( 1 - 4.39e17T + 1.30e35T^{2} \)
53 \( 1 + 2.38e18T + 1.62e36T^{2} \)
59 \( 1 + 1.88e18iT - 1.54e37T^{2} \)
61 \( 1 - 6.78e18iT - 3.10e37T^{2} \)
67 \( 1 - 2.14e19T + 2.22e38T^{2} \)
71 \( 1 - 8.96e16T + 7.52e38T^{2} \)
73 \( 1 + 3.12e19T + 1.34e39T^{2} \)
79 \( 1 + 1.16e19iT - 7.08e39T^{2} \)
83 \( 1 - 1.13e20iT - 1.99e40T^{2} \)
89 \( 1 + 4.92e20iT - 8.65e40T^{2} \)
97 \( 1 - 1.78e20T + 5.27e41T^{2} \)
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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

−11.33486343226147474804639262682, −9.940314020187948683252962689485, −9.417190645343170741760977820699, −8.111401134371966963560968080112, −7.25861840666026470858359880265, −5.93396361798908531552733030989, −4.31491214099468538129036392818, −3.41775561929985103327188799407, −2.06539055122131033291284797277, −1.33679154628391176154037571291, 0.009165362873881211436331854709, 0.907438004731568445182896903555, 2.10964404310622474795986940241, 3.46328206919372611031038053891, 5.21789253163012086566465975554, 5.75990660926528994558245000903, 7.22908536781700030053271655740, 7.87043594329217179183823546193, 9.155834662579199421291757520215, 9.904069021272908229016473237325

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