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Properties

Label 2-72-24.11-c3-0-4
Degree $2$
Conductor $72$
Sign $0.102 - 0.994i$
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  + (−0.977 + 2.65i)2-s + (−6.08 − 5.18i)4-s + 19.8·5-s + 12.9i·7-s + (19.7 − 11.0i)8-s + (−19.4 + 52.7i)10-s + 52.8i·11-s − 24.4i·13-s + (−34.2 − 12.6i)14-s + (10.1 + 63.1i)16-s + 25.0i·17-s − 37.3·19-s + (−121. − 103. i)20-s + (−140. − 51.6i)22-s + 166.·23-s + ⋯
L(s)  = 1  + (−0.345 + 0.938i)2-s + (−0.761 − 0.648i)4-s + 1.77·5-s + 0.696i·7-s + (0.871 − 0.490i)8-s + (−0.614 + 1.66i)10-s + 1.44i·11-s − 0.521i·13-s + (−0.653 − 0.240i)14-s + (0.158 + 0.987i)16-s + 0.356i·17-s − 0.451·19-s + (−1.35 − 1.15i)20-s + (−1.35 − 0.500i)22-s + 1.51·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.102 - 0.994i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ 0.102 - 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.10025 + 0.992215i\)
\(L(\frac12)\) \(\approx\) \(1.10025 + 0.992215i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.977 - 2.65i)T \)
3 \( 1 \)
good5 \( 1 - 19.8T + 125T^{2} \)
7 \( 1 - 12.9iT - 343T^{2} \)
11 \( 1 - 52.8iT - 1.33e3T^{2} \)
13 \( 1 + 24.4iT - 2.19e3T^{2} \)
17 \( 1 - 25.0iT - 4.91e3T^{2} \)
19 \( 1 + 37.3T + 6.85e3T^{2} \)
23 \( 1 - 166.T + 1.21e4T^{2} \)
29 \( 1 + 81.1T + 2.43e4T^{2} \)
31 \( 1 + 149. iT - 2.97e4T^{2} \)
37 \( 1 + 244. iT - 5.06e4T^{2} \)
41 \( 1 + 356. iT - 6.89e4T^{2} \)
43 \( 1 + 400.T + 7.95e4T^{2} \)
47 \( 1 + 360.T + 1.03e5T^{2} \)
53 \( 1 - 32.8T + 1.48e5T^{2} \)
59 \( 1 + 82.9iT - 2.05e5T^{2} \)
61 \( 1 - 299. iT - 2.26e5T^{2} \)
67 \( 1 + 544.T + 3.00e5T^{2} \)
71 \( 1 - 576.T + 3.57e5T^{2} \)
73 \( 1 + 361.T + 3.89e5T^{2} \)
79 \( 1 + 1.20e3iT - 4.93e5T^{2} \)
83 \( 1 + 272. iT - 5.71e5T^{2} \)
89 \( 1 - 682. iT - 7.04e5T^{2} \)
97 \( 1 - 248.T + 9.12e5T^{2} \)
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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

−14.71602722820338371435122392347, −13.38078911794291264196360953732, −12.69822960712386469579589712529, −10.52845614189943751169504561286, −9.633787190484800268810938463322, −8.836653534965852817392905160925, −7.14177039222427380488149473341, −5.96071957913901424008467436430, −5.01666204752134355957320329726, −1.95955659372486566174527526639, 1.31762582779671920723573005782, 3.04464913947342352187069413951, 5.06198957116107880284405155447, 6.62550581566158733034998076670, 8.555838418873659235070542788476, 9.515912169666991914481019848856, 10.48931271703938715454297787555, 11.37098214572509183038712652975, 13.09198735677872772962771341045, 13.55032931614013428352220176580

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