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Properties

Label 2-24-8.3-c2-0-2
Degree $2$
Conductor $24$
Sign $0.524 + 0.851i$
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  + (−0.366 − 1.96i)2-s + 1.73·3-s + (−3.73 + 1.43i)4-s − 2.87i·5-s + (−0.633 − 3.40i)6-s + 10.7i·7-s + (4.19 + 6.81i)8-s + 2.99·9-s + (−5.66 + 1.05i)10-s − 8·11-s + (−6.46 + 2.49i)12-s − 15.7i·13-s + (21.1 − 3.93i)14-s − 4.98i·15-s + (11.8 − 10.7i)16-s − 15.8·17-s + ⋯
L(s)  = 1  + (−0.183 − 0.983i)2-s + 0.577·3-s + (−0.933 + 0.359i)4-s − 0.575i·5-s + (−0.105 − 0.567i)6-s + 1.53i·7-s + (0.524 + 0.851i)8-s + 0.333·9-s + (−0.566 + 0.105i)10-s − 0.727·11-s + (−0.538 + 0.207i)12-s − 1.20i·13-s + (1.50 − 0.280i)14-s − 0.332i·15-s + (0.741 − 0.671i)16-s − 0.932·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.524 + 0.851i$
Analytic conductor: \(0.653952\)
Root analytic conductor: \(0.808673\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :1),\ 0.524 + 0.851i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.784597 - 0.438174i\)
\(L(\frac12)\) \(\approx\) \(0.784597 - 0.438174i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.96i)T \)
3 \( 1 - 1.73T \)
good5 \( 1 + 2.87iT - 25T^{2} \)
7 \( 1 - 10.7iT - 49T^{2} \)
11 \( 1 + 8T + 121T^{2} \)
13 \( 1 + 15.7iT - 169T^{2} \)
17 \( 1 + 15.8T + 289T^{2} \)
19 \( 1 - 1.07T + 361T^{2} \)
23 \( 1 + 21.4iT - 529T^{2} \)
29 \( 1 - 40.0iT - 841T^{2} \)
31 \( 1 - 9.20iT - 961T^{2} \)
37 \( 1 + 9.97iT - 1.36e3T^{2} \)
41 \( 1 - 51.5T + 1.68e3T^{2} \)
43 \( 1 + 12.7T + 1.84e3T^{2} \)
47 \( 1 - 1.54iT - 2.20e3T^{2} \)
53 \( 1 + 28.5iT - 2.80e3T^{2} \)
59 \( 1 + 11.2T + 3.48e3T^{2} \)
61 \( 1 + 1.54iT - 3.72e3T^{2} \)
67 \( 1 + 43.2T + 4.48e3T^{2} \)
71 \( 1 - 84.4iT - 5.04e3T^{2} \)
73 \( 1 - 105.T + 5.32e3T^{2} \)
79 \( 1 + 73.6iT - 6.24e3T^{2} \)
83 \( 1 - 12.2T + 6.88e3T^{2} \)
89 \( 1 - 33.1T + 7.92e3T^{2} \)
97 \( 1 + 69.1T + 9.40e3T^{2} \)
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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

−17.86894800987277571094855865699, −16.00850540241678952529086997517, −14.74469057313655704085639288379, −13.01856626614660602989844745783, −12.40726944006768489398401158527, −10.70464897985039189470434307052, −9.110724250132546308985061125581, −8.293193361109677564047267773112, −5.14059384744607714226052879422, −2.68155907806404530091965479096, 4.24261628617026933892896956709, 6.74789168780566424944579060693, 7.77757615597159208351556357411, 9.496348007137509508991465712168, 10.80626742570984102236577363067, 13.38007144419951117460720073910, 13.99366662794992606072480698545, 15.23961486911368683205397555358, 16.45267067245767059072561184582, 17.57154116753396557950094918534

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