Properties

Label 2-72-72.29-c2-0-16
Degree $2$
Conductor $72$
Sign $0.997 + 0.0715i$
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  + (1.96 − 0.355i)2-s + (2.21 + 2.02i)3-s + (3.74 − 1.39i)4-s + (−4.28 − 7.41i)5-s + (5.07 + 3.19i)6-s + (−3.75 + 6.50i)7-s + (6.87 − 4.08i)8-s + (0.811 + 8.96i)9-s + (−11.0 − 13.0i)10-s + (−4.74 + 8.22i)11-s + (11.1 + 4.48i)12-s + (−9.54 + 5.51i)13-s + (−5.08 + 14.1i)14-s + (5.52 − 25.0i)15-s + (12.0 − 10.4i)16-s − 11.3i·17-s + ⋯
L(s)  = 1  + (0.984 − 0.177i)2-s + (0.738 + 0.674i)3-s + (0.936 − 0.349i)4-s + (−0.856 − 1.48i)5-s + (0.846 + 0.532i)6-s + (−0.536 + 0.929i)7-s + (0.859 − 0.510i)8-s + (0.0901 + 0.995i)9-s + (−1.10 − 1.30i)10-s + (−0.431 + 0.747i)11-s + (0.927 + 0.373i)12-s + (−0.734 + 0.424i)13-s + (−0.362 + 1.01i)14-s + (0.368 − 1.67i)15-s + (0.755 − 0.655i)16-s − 0.667i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0715i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.997 + 0.0715i$
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: \((2,\ 72,\ (\ :1),\ 0.997 + 0.0715i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.19074 - 0.0784915i\)
\(L(\frac12)\) \(\approx\) \(2.19074 - 0.0784915i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.96 + 0.355i)T \)
3 \( 1 + (-2.21 - 2.02i)T \)
good5 \( 1 + (4.28 + 7.41i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (3.75 - 6.50i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.74 - 8.22i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (9.54 - 5.51i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 11.3iT - 289T^{2} \)
19 \( 1 + 18.3iT - 361T^{2} \)
23 \( 1 + (-22.8 + 13.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-3.48 + 6.03i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-6.42 - 11.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 5.89iT - 1.36e3T^{2} \)
41 \( 1 + (-32.7 + 18.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-21.1 - 12.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (15.8 + 9.17i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 58.4T + 2.80e3T^{2} \)
59 \( 1 + (13.1 + 22.8i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-56.0 - 32.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-28.4 + 16.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 84.5iT - 5.04e3T^{2} \)
73 \( 1 + 94.1T + 5.32e3T^{2} \)
79 \( 1 + (12.3 - 21.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (57.7 - 100. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 + (-94.1 + 163. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−14.48346620813422901012644877062, −13.05700281641711395398704639365, −12.48761412303616192217774988943, −11.40270017353535162387047195938, −9.716249825308382769260440361329, −8.779709980671238895844470007019, −7.34330681930960900547621849691, −5.12451415488169425331454456871, −4.45867241592984829968881067654, −2.68536446137902228647628924952, 2.93372240570641062497714866881, 3.75599078979891668462052104535, 6.25337164494714271673155703185, 7.29040700556018510391367955757, 7.913738166811810337349433374131, 10.27350345851459437189731302347, 11.24261033113268934282375612348, 12.50820532464063687939225596340, 13.47863645096364538077723393391, 14.43279827691132378456998261288

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