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Properties

Label 2-72-24.5-c4-0-0
Degree $2$
Conductor $72$
Sign $-0.792 + 0.609i$
Analytic cond. $7.44263$
Root an. cond. $2.72811$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.64 + 3.64i)2-s + (−10.5 + 11.9i)4-s − 21.8·5-s + 2.75·7-s + (−61.1 − 18.9i)8-s + (−35.9 − 79.6i)10-s − 173.·11-s − 121. i·13-s + (4.52 + 10.0i)14-s + (−31.5 − 254. i)16-s + 347. i·17-s + 88.8i·19-s + (231. − 261. i)20-s + (−285. − 633. i)22-s + 230. i·23-s + ⋯
L(s)  = 1  + (0.411 + 0.911i)2-s + (−0.662 + 0.749i)4-s − 0.873·5-s + 0.0561·7-s + (−0.955 − 0.295i)8-s + (−0.359 − 0.796i)10-s − 1.43·11-s − 0.721i·13-s + (0.0230 + 0.0511i)14-s + (−0.123 − 0.992i)16-s + 1.20i·17-s + 0.246i·19-s + (0.578 − 0.654i)20-s + (−0.590 − 1.30i)22-s + 0.436i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.792 + 0.609i$
Analytic conductor: \(7.44263\)
Root analytic conductor: \(2.72811\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :2),\ -0.792 + 0.609i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.152714 - 0.449295i\)
\(L(\frac12)\) \(\approx\) \(0.152714 - 0.449295i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.64 - 3.64i)T \)
3 \( 1 \)
good5 \( 1 + 21.8T + 625T^{2} \)
7 \( 1 - 2.75T + 2.40e3T^{2} \)
11 \( 1 + 173.T + 1.46e4T^{2} \)
13 \( 1 + 121. iT - 2.85e4T^{2} \)
17 \( 1 - 347. iT - 8.35e4T^{2} \)
19 \( 1 - 88.8iT - 1.30e5T^{2} \)
23 \( 1 - 230. iT - 2.79e5T^{2} \)
29 \( 1 + 1.31e3T + 7.07e5T^{2} \)
31 \( 1 - 1.67e3T + 9.23e5T^{2} \)
37 \( 1 - 1.40e3iT - 1.87e6T^{2} \)
41 \( 1 - 2.07e3iT - 2.82e6T^{2} \)
43 \( 1 - 3.31e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.47e3iT - 4.87e6T^{2} \)
53 \( 1 - 579.T + 7.89e6T^{2} \)
59 \( 1 - 2.27e3T + 1.21e7T^{2} \)
61 \( 1 + 5.79e3iT - 1.38e7T^{2} \)
67 \( 1 + 2.80e3iT - 2.01e7T^{2} \)
71 \( 1 + 7.67e3iT - 2.54e7T^{2} \)
73 \( 1 + 8.36e3T + 2.83e7T^{2} \)
79 \( 1 - 1.01e3T + 3.89e7T^{2} \)
83 \( 1 - 305.T + 4.74e7T^{2} \)
89 \( 1 + 1.94e3iT - 6.27e7T^{2} \)
97 \( 1 + 7.67e3T + 8.85e7T^{2} \)
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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

−14.93808131508743217273316911123, −13.39287906153429823430058486527, −12.71223711878559131121383554035, −11.42572349647126765241404324483, −9.977817001450070681593776674069, −8.153955789168717193349541100234, −7.80101673864122612253586573253, −6.11496841790381336208261214033, −4.78758776286810002850183004184, −3.30984354869934721500138517098, 0.20992042405445104794256612375, 2.54358396823476600768049226270, 4.12912012051496936497816895712, 5.39624670068343400606659951126, 7.35639891449711747626393444414, 8.764738546917228436149106294740, 10.09132411076119579420488709710, 11.21511274949741262109865217617, 11.99255497678330135316046588315, 13.13286207517023139438004258628

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