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Properties

Label 2-24-8.5-c3-0-3
Degree $2$
Conductor $24$
Sign $0.970 - 0.242i$
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  + (2.55 + 1.20i)2-s − 3i·3-s + (5.07 + 6.18i)4-s − 0.612i·5-s + (3.62 − 7.66i)6-s − 22.7·7-s + (5.48 + 21.9i)8-s − 9·9-s + (0.741 − 1.56i)10-s − 60.2i·11-s + (18.5 − 15.2i)12-s + 52.9i·13-s + (−58.1 − 27.5i)14-s − 1.83·15-s + (−12.5 + 62.7i)16-s + 47.1·17-s + ⋯
L(s)  = 1  + (0.903 + 0.427i)2-s − 0.577i·3-s + (0.634 + 0.773i)4-s − 0.0547i·5-s + (0.246 − 0.521i)6-s − 1.22·7-s + (0.242 + 0.970i)8-s − 0.333·9-s + (0.0234 − 0.0495i)10-s − 1.65i·11-s + (0.446 − 0.366i)12-s + 1.12i·13-s + (−1.11 − 0.525i)14-s − 0.0316·15-s + (−0.195 + 0.980i)16-s + 0.672·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.970 - 0.242i$
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: \((2,\ 24,\ (\ :3/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.62966 + 0.200522i\)
\(L(\frac12)\) \(\approx\) \(1.62966 + 0.200522i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.55 - 1.20i)T \)
3 \( 1 + 3iT \)
good5 \( 1 + 0.612iT - 125T^{2} \)
7 \( 1 + 22.7T + 343T^{2} \)
11 \( 1 + 60.2iT - 1.33e3T^{2} \)
13 \( 1 - 52.9iT - 2.19e3T^{2} \)
17 \( 1 - 47.1T + 4.91e3T^{2} \)
19 \( 1 - 29.1iT - 6.85e3T^{2} \)
23 \( 1 - 109.T + 1.21e4T^{2} \)
29 \( 1 + 10.4iT - 2.43e4T^{2} \)
31 \( 1 + 220.T + 2.97e4T^{2} \)
37 \( 1 + 408. iT - 5.06e4T^{2} \)
41 \( 1 + 360.T + 6.89e4T^{2} \)
43 \( 1 - 236. iT - 7.95e4T^{2} \)
47 \( 1 - 129.T + 1.03e5T^{2} \)
53 \( 1 + 117. iT - 1.48e5T^{2} \)
59 \( 1 - 262. iT - 2.05e5T^{2} \)
61 \( 1 - 273. iT - 2.26e5T^{2} \)
67 \( 1 - 89.4iT - 3.00e5T^{2} \)
71 \( 1 + 350.T + 3.57e5T^{2} \)
73 \( 1 - 532.T + 3.89e5T^{2} \)
79 \( 1 + 166.T + 4.93e5T^{2} \)
83 \( 1 + 361. iT - 5.71e5T^{2} \)
89 \( 1 - 40.3T + 7.04e5T^{2} \)
97 \( 1 + 614.T + 9.12e5T^{2} \)
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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

−16.66014789458222637111423026697, −16.29940376926440370273287011641, −14.51054383651718801644908773311, −13.47661527433415316794602012673, −12.50480788631677223728495807123, −11.14522382083617816170406917475, −8.830876638050946609588317383171, −7.02705245887260432992820720157, −5.82811196670478257227218184066, −3.33245581226739533820022894569, 3.20581628340100797581935383163, 5.09262476471760179754107721098, 6.90466347330710146688386266160, 9.627253267756833406486436427218, 10.52949642652849951871680281109, 12.30463154095035302554872658967, 13.14567633999045495945887075924, 14.86088823567976231364516897584, 15.53235172005544875272043666806, 16.90144288421934085957714039741

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