Properties

Degree $2$
Conductor $3024$
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

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.0288 + 0.999i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.0288 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.647112658\)
\(L(\frac12)\) \(\approx\) \(1.647112658\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.0763i)T \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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