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Properties

Label 2-24-24.5-c2-0-4
Degree $2$
Conductor $24$
Sign $0.829 + 0.559i$
Analytic cond. $0.653952$
Root an. cond. $0.808673$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.87i)2-s + (1.41 + 2.64i)3-s + (−3 − 2.64i)4-s − 5.65·5-s + (5.94 − 0.774i)6-s + 4·7-s + (−7.07 + 3.74i)8-s + (−5 + 7.48i)9-s + (−4.00 + 10.5i)10-s + 8.48·11-s + (2.75 − 11.6i)12-s − 10.5i·13-s + (2.82 − 7.48i)14-s + (−8.00 − 14.9i)15-s + (1.99 + 15.8i)16-s − 14.9i·17-s + ⋯
L(s)  = 1  + (0.353 − 0.935i)2-s + (0.471 + 0.881i)3-s + (−0.750 − 0.661i)4-s − 1.13·5-s + (0.991 − 0.129i)6-s + 0.571·7-s + (−0.883 + 0.467i)8-s + (−0.555 + 0.831i)9-s + (−0.400 + 1.05i)10-s + 0.771·11-s + (0.229 − 0.973i)12-s − 0.814i·13-s + (0.202 − 0.534i)14-s + (−0.533 − 0.997i)15-s + (0.124 + 0.992i)16-s − 0.880i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.829 + 0.559i$
Analytic conductor: \(0.653952\)
Root analytic conductor: \(0.808673\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :1),\ 0.829 + 0.559i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.982034 - 0.300134i\)
\(L(\frac12)\) \(\approx\) \(0.982034 - 0.300134i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.87i)T \)
3 \( 1 + (-1.41 - 2.64i)T \)
good5 \( 1 + 5.65T + 25T^{2} \)
7 \( 1 - 4T + 49T^{2} \)
11 \( 1 - 8.48T + 121T^{2} \)
13 \( 1 + 10.5iT - 169T^{2} \)
17 \( 1 + 14.9iT - 289T^{2} \)
19 \( 1 + 5.29iT - 361T^{2} \)
23 \( 1 - 29.9iT - 529T^{2} \)
29 \( 1 - 16.9T + 841T^{2} \)
31 \( 1 + 4T + 961T^{2} \)
37 \( 1 - 52.9iT - 1.36e3T^{2} \)
41 \( 1 + 29.9iT - 1.68e3T^{2} \)
43 \( 1 - 5.29iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 50.9T + 2.80e3T^{2} \)
59 \( 1 - 48.0T + 3.48e3T^{2} \)
61 \( 1 + 95.2iT - 3.72e3T^{2} \)
67 \( 1 - 47.6iT - 4.48e3T^{2} \)
71 \( 1 + 89.7iT - 5.04e3T^{2} \)
73 \( 1 + 6T + 5.32e3T^{2} \)
79 \( 1 - 124T + 6.24e3T^{2} \)
83 \( 1 + 2.82T + 6.88e3T^{2} \)
89 \( 1 - 104. iT - 7.92e3T^{2} \)
97 \( 1 - 118T + 9.40e3T^{2} \)
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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

−17.54059358462985113132075134316, −15.75800154516955348357620109927, −14.87402065275773677012619694922, −13.70463323740235615384453330556, −11.89746978123167618675420931678, −11.04438544989209094837587220671, −9.544275704799949303938687941939, −8.122046395595495303490504147548, −4.89793294545700987547852631759, −3.45864190635624545873021523023, 4.05866501531734767507822894176, 6.52670727400072296712392964723, 7.82115244114720862878575879135, 8.798698392185051621009315897507, 11.67471531999220073326774440554, 12.65300169127315327592621779684, 14.22533793456443202711233367020, 14.86990152471455953834648148027, 16.32287850430958827192534205426, 17.55902406059585357294671813029

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