Properties

Label 2-24-8.5-c3-0-2
Degree $2$
Conductor $24$
Sign $0.339 + 0.940i$
Analytic cond. $1.41604$
Root an. cond. $1.18997$
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  + (−2.80 + 0.325i)2-s − 3i·3-s + (7.78 − 1.83i)4-s − 18.5i·5-s + (0.977 + 8.42i)6-s + 9.32·7-s + (−21.2 + 7.68i)8-s − 9·9-s + (6.04 + 52.0i)10-s + 39.7i·11-s + (−5.49 − 23.3i)12-s − 32.9i·13-s + (−26.2 + 3.04i)14-s − 55.6·15-s + (57.2 − 28.5i)16-s + 90.5·17-s + ⋯
L(s)  = 1  + (−0.993 + 0.115i)2-s − 0.577i·3-s + (0.973 − 0.228i)4-s − 1.65i·5-s + (0.0665 + 0.573i)6-s + 0.503·7-s + (−0.940 + 0.339i)8-s − 0.333·9-s + (0.191 + 1.64i)10-s + 1.08i·11-s + (−0.132 − 0.562i)12-s − 0.703i·13-s + (−0.500 + 0.0580i)14-s − 0.957·15-s + (0.895 − 0.445i)16-s + 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.339 + 0.940i$
Analytic conductor: \(1.41604\)
Root analytic conductor: \(1.18997\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :3/2),\ 0.339 + 0.940i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.636769 - 0.447083i\)
\(L(\frac12)\) \(\approx\) \(0.636769 - 0.447083i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.80 - 0.325i)T \)
3 \( 1 + 3iT \)
good5 \( 1 + 18.5iT - 125T^{2} \)
7 \( 1 - 9.32T + 343T^{2} \)
11 \( 1 - 39.7iT - 1.33e3T^{2} \)
13 \( 1 + 32.9iT - 2.19e3T^{2} \)
17 \( 1 - 90.5T + 4.91e3T^{2} \)
19 \( 1 - 72.5iT - 6.85e3T^{2} \)
23 \( 1 - 45.3T + 1.21e4T^{2} \)
29 \( 1 - 143. iT - 2.43e4T^{2} \)
31 \( 1 - 90.4T + 2.97e4T^{2} \)
37 \( 1 + 1.77iT - 5.06e4T^{2} \)
41 \( 1 - 195.T + 6.89e4T^{2} \)
43 \( 1 + 407. iT - 7.95e4T^{2} \)
47 \( 1 + 278.T + 1.03e5T^{2} \)
53 \( 1 + 241. iT - 1.48e5T^{2} \)
59 \( 1 - 149. iT - 2.05e5T^{2} \)
61 \( 1 - 508. iT - 2.26e5T^{2} \)
67 \( 1 - 950. iT - 3.00e5T^{2} \)
71 \( 1 + 803.T + 3.57e5T^{2} \)
73 \( 1 - 449.T + 3.89e5T^{2} \)
79 \( 1 + 157.T + 4.93e5T^{2} \)
83 \( 1 + 175. iT - 5.71e5T^{2} \)
89 \( 1 - 127.T + 7.04e5T^{2} \)
97 \( 1 - 158.T + 9.12e5T^{2} \)
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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

−17.14476168075069414521358555668, −16.19794799803469453426678271317, −14.71019484089938583453067523721, −12.73978053989557095084042727127, −11.96299671768611326057438051463, −10.01859238145999464636812286335, −8.605645948350586607916413881782, −7.59343552523933760602606176166, −5.39861801376524776329793114427, −1.32506279373975630882762425965, 3.05990531721135832667930574220, 6.34884103545844015933093148626, 7.895849629935423211689236915711, 9.605669855423760745739101161610, 10.87213452841818450821183791433, 11.51053377909254462772901100762, 14.13831872233514156441667605230, 15.11585186432710096099572868342, 16.36636102613912267939657393564, 17.61624454576484316199602628817

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