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Properties

Label 2-24-8.5-c3-0-5
Degree $2$
Conductor $24$
Sign $0.192 + 0.981i$
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  + (1.25 − 2.53i)2-s − 3i·3-s + (−4.86 − 6.35i)4-s + 9.15i·5-s + (−7.60 − 3.75i)6-s + 27.4·7-s + (−22.2 + 4.36i)8-s − 9·9-s + (23.2 + 11.4i)10-s + 20.5i·11-s + (−19.0 + 14.5i)12-s + 32.0i·13-s + (34.3 − 69.5i)14-s + 27.4·15-s + (−16.7 + 61.7i)16-s − 111.·17-s + ⋯
L(s)  = 1  + (0.442 − 0.896i)2-s − 0.577i·3-s + (−0.607 − 0.794i)4-s + 0.818i·5-s + (−0.517 − 0.255i)6-s + 1.48·7-s + (−0.981 + 0.192i)8-s − 0.333·9-s + (0.734 + 0.362i)10-s + 0.562i·11-s + (−0.458 + 0.350i)12-s + 0.683i·13-s + (0.655 − 1.32i)14-s + 0.472·15-s + (−0.261 + 0.965i)16-s − 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.192 + 0.981i$
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: \((2,\ 24,\ (\ :3/2),\ 0.192 + 0.981i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.05857 - 0.870807i\)
\(L(\frac12)\) \(\approx\) \(1.05857 - 0.870807i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 + 2.53i)T \)
3 \( 1 + 3iT \)
good5 \( 1 - 9.15iT - 125T^{2} \)
7 \( 1 - 27.4T + 343T^{2} \)
11 \( 1 - 20.5iT - 1.33e3T^{2} \)
13 \( 1 - 32.0iT - 2.19e3T^{2} \)
17 \( 1 + 111.T + 4.91e3T^{2} \)
19 \( 1 + 129. iT - 6.85e3T^{2} \)
23 \( 1 - 9.16T + 1.21e4T^{2} \)
29 \( 1 - 41.0iT - 2.43e4T^{2} \)
31 \( 1 + 187.T + 2.97e4T^{2} \)
37 \( 1 - 114. iT - 5.06e4T^{2} \)
41 \( 1 - 282.T + 6.89e4T^{2} \)
43 \( 1 + 89.3iT - 7.95e4T^{2} \)
47 \( 1 + 54.6T + 1.03e5T^{2} \)
53 \( 1 + 726. iT - 1.48e5T^{2} \)
59 \( 1 + 216. iT - 2.05e5T^{2} \)
61 \( 1 - 754. iT - 2.26e5T^{2} \)
67 \( 1 + 379. iT - 3.00e5T^{2} \)
71 \( 1 - 302.T + 3.57e5T^{2} \)
73 \( 1 + 504.T + 3.89e5T^{2} \)
79 \( 1 - 301.T + 4.93e5T^{2} \)
83 \( 1 + 599. iT - 5.71e5T^{2} \)
89 \( 1 + 277.T + 7.04e5T^{2} \)
97 \( 1 + 765.T + 9.12e5T^{2} \)
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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

−17.70141927153466442496081440605, −15.12554879465913182870736894004, −14.29404723064282305545418114153, −13.12798322866584959052141297608, −11.51497629264453074409614985873, −10.92415371421557910430604356644, −8.910819404006661901345732694658, −6.91322891317605065833849060303, −4.72169953520010168620514549605, −2.16303243989996515704103791579, 4.35143178292738007343162123873, 5.59776002128949147467911890689, 7.957427606768841897834892410294, 8.923402948061377104839545913169, 11.07528357697740631256886944908, 12.61623161062397753039932514790, 14.03849563937045367865012755845, 15.05179427976862094357557660672, 16.20885536902406643947887304763, 17.21357036204838171694064133849

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