Properties

Degree $2$
Conductor $3024$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.46i·5-s + 2.64·7-s − 4.46i·11-s + 7.34i·17-s − 8.64·19-s − 2.88i·23-s − 14.9·25-s − 8.29·31-s + 11.8i·35-s − 3.93·37-s + 2.88i·41-s + 7.00·49-s + 19.9·55-s + 10.2i·71-s − 11.8i·77-s + ⋯
L(s)  = 1  + 1.99i·5-s + 0.999·7-s − 1.34i·11-s + 1.78i·17-s − 1.98·19-s − 0.601i·23-s − 2.98·25-s − 1.48·31-s + 1.99i·35-s − 0.647·37-s + 0.450i·41-s + 49-s + 2.68·55-s + 1.21i·71-s − 1.34i·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8972819903\)
\(L(\frac12)\) \(\approx\) \(0.8972819903\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 2.64T \)
good5 \( 1 - 4.46iT - 5T^{2} \)
11 \( 1 + 4.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 7.34iT - 17T^{2} \)
19 \( 1 + 8.64T + 19T^{2} \)
23 \( 1 + 2.88iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.29T + 31T^{2} \)
37 \( 1 + 3.93T + 37T^{2} \)
41 \( 1 - 2.88iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 14.9iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.843376895432560156434243785953, −8.291680676290566892053322968197, −7.65959884368767755289767480168, −6.66388610303957964213708399664, −6.24085836357607020114012344346, −5.52446079690529930841310761948, −4.09404699640716328795613414980, −3.62489446604934505372337612264, −2.53810728575792409064889053264, −1.79070178846138437458027219830, 0.25751857565385617252215911255, 1.59564993816268375051735941066, 2.14658754433232075816287952598, 3.93958572650166429128345566939, 4.67573949519844408537769600837, 4.98863158800812318920988707241, 5.77322219928957286700049287249, 7.08743612439311620998034489680, 7.67104865278592566441094040829, 8.467042656768785697382189044367

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