Properties

Label 2-24-8.5-c7-0-13
Degree $2$
Conductor $24$
Sign $-0.769 + 0.638i$
Analytic cond. $7.49724$
Root an. cond. $2.73810$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.24 − 7.74i)2-s − 27i·3-s + (7.88 − 127. i)4-s − 76.0i·5-s + (−209. − 222. i)6-s − 222.·7-s + (−925. − 1.11e3i)8-s − 729·9-s + (−589. − 627. i)10-s − 1.90e3i·11-s + (−3.44e3 − 212. i)12-s + 309. i·13-s + (−1.83e3 + 1.72e3i)14-s − 2.05e3·15-s + (−1.62e4 − 2.01e3i)16-s + 1.67e4·17-s + ⋯
L(s)  = 1  + (0.728 − 0.684i)2-s − 0.577i·3-s + (0.0615 − 0.998i)4-s − 0.272i·5-s + (−0.395 − 0.420i)6-s − 0.245·7-s + (−0.638 − 0.769i)8-s − 0.333·9-s + (−0.186 − 0.198i)10-s − 0.431i·11-s + (−0.576 − 0.0355i)12-s + 0.0391i·13-s + (−0.178 + 0.168i)14-s − 0.157·15-s + (−0.992 − 0.122i)16-s + 0.826·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $-0.769 + 0.638i$
Analytic conductor: \(7.49724\)
Root analytic conductor: \(2.73810\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :7/2),\ -0.769 + 0.638i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.732360 - 2.02843i\)
\(L(\frac12)\) \(\approx\) \(0.732360 - 2.02843i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-8.24 + 7.74i)T \)
3 \( 1 + 27iT \)
good5 \( 1 + 76.0iT - 7.81e4T^{2} \)
7 \( 1 + 222.T + 8.23e5T^{2} \)
11 \( 1 + 1.90e3iT - 1.94e7T^{2} \)
13 \( 1 - 309. iT - 6.27e7T^{2} \)
17 \( 1 - 1.67e4T + 4.10e8T^{2} \)
19 \( 1 + 2.70e4iT - 8.93e8T^{2} \)
23 \( 1 - 7.73e4T + 3.40e9T^{2} \)
29 \( 1 - 1.56e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.65e5T + 2.75e10T^{2} \)
37 \( 1 + 1.13e5iT - 9.49e10T^{2} \)
41 \( 1 + 6.94e5T + 1.94e11T^{2} \)
43 \( 1 + 9.00e5iT - 2.71e11T^{2} \)
47 \( 1 + 7.71e4T + 5.06e11T^{2} \)
53 \( 1 - 1.89e6iT - 1.17e12T^{2} \)
59 \( 1 - 7.04e5iT - 2.48e12T^{2} \)
61 \( 1 - 1.42e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.87e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.31e6T + 9.09e12T^{2} \)
73 \( 1 + 3.90e4T + 1.10e13T^{2} \)
79 \( 1 + 2.43e6T + 1.92e13T^{2} \)
83 \( 1 - 6.00e6iT - 2.71e13T^{2} \)
89 \( 1 - 2.38e6T + 4.42e13T^{2} \)
97 \( 1 - 1.31e7T + 8.07e13T^{2} \)
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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

−15.35606080850383664252390644179, −14.01745885278633411378496809195, −13.00262248037900166477402000082, −11.95873605428012612020039669138, −10.62219293049180299787938641197, −8.933657906152764671209435215789, −6.78173547852589269572097773862, −5.13318978617244759995592732136, −3.02827804385519465888236066080, −0.983594728582720254391568536215, 3.20152666435461835305238034415, 4.88492362005164650725829898562, 6.50887188387878640469354715852, 8.152344378713790839596574461447, 9.906826576083317500163436607658, 11.62465402207951116844687678563, 12.97370565622669055500744623183, 14.38242803872315937288716527160, 15.24274631313763369764915410348, 16.41799030271473782775488860925

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