Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.0288 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.10·5-s + (2.64 − 0.0763i)7-s − 3.42i·11-s + 4.98i·13-s − 6.38·17-s + 4.65i·19-s + 8.98i·23-s − 3.77·25-s + 1.51i·29-s + 6.71i·31-s + (2.92 − 0.0843i)35-s − 2.83·37-s − 10.1·41-s − 10.8·43-s + 12.8·47-s + ⋯
L(s)  = 1  + 0.494·5-s + (0.999 − 0.0288i)7-s − 1.03i·11-s + 1.38i·13-s − 1.54·17-s + 1.06i·19-s + 1.87i·23-s − 0.755·25-s + 0.280i·29-s + 1.20i·31-s + (0.494 − 0.0142i)35-s − 0.465·37-s − 1.57·41-s − 1.66·43-s + 1.88·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0288 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.0288 - 0.999i)\)
\(L(1)\)  \(\approx\)  \(1.647112658\)
\(L(\frac12)\)  \(\approx\)  \(1.647112658\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.64 + 0.0763i)T \)
good5 \( 1 - 1.10T + 5T^{2} \)
11 \( 1 + 3.42iT - 11T^{2} \)
13 \( 1 - 4.98iT - 13T^{2} \)
17 \( 1 + 6.38T + 17T^{2} \)
19 \( 1 - 4.65iT - 19T^{2} \)
23 \( 1 - 8.98iT - 23T^{2} \)
29 \( 1 - 1.51iT - 29T^{2} \)
31 \( 1 - 6.71iT - 31T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 - 3.02iT - 53T^{2} \)
59 \( 1 - 9.54T + 59T^{2} \)
61 \( 1 - 9.16iT - 61T^{2} \)
67 \( 1 - 5.07T + 67T^{2} \)
71 \( 1 + 8.94iT - 71T^{2} \)
73 \( 1 - 7.79iT - 73T^{2} \)
79 \( 1 - 2.05T + 79T^{2} \)
83 \( 1 - 5.73T + 83T^{2} \)
89 \( 1 - 6.47T + 89T^{2} \)
97 \( 1 + 8.04iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.720072560835751676349697793515, −8.440372082443741587118639957114, −7.32993297443537037010097852791, −6.70079463729770677220943251679, −5.77597535536724257607348827958, −5.17311023573092682743021847032, −4.19959423203676710210192829656, −3.44117596734134171987230375884, −2.02097922892590886727079751597, −1.50858773493573608121928363034, 0.48189226895012531539946381569, 2.06276403116188586314845361161, 2.45405578209301395783719845691, 3.93805460273500290499071356335, 4.80415094642060948130911985520, 5.24141691403327654462842218851, 6.36580748996097746517628898210, 6.96603758636628561895334201174, 7.894597079128761755835390319191, 8.489759147853978058959185129900

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