Properties

Degree $2$
Conductor $3024$
Sign $0.391 + 0.920i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.86·5-s + (−2.43 + 1.03i)7-s + 3.32i·11-s + 0.821i·13-s − 3.52·17-s − 5.25i·19-s + 5.12i·23-s + 3.22·25-s + 1.64i·29-s + 2.55i·31-s + (6.98 − 2.96i)35-s − 8.93·37-s − 3.49·41-s + 0.161·43-s + 5.34·47-s + ⋯
L(s)  = 1  − 1.28·5-s + (−0.920 + 0.391i)7-s + 1.00i·11-s + 0.227i·13-s − 0.853·17-s − 1.20i·19-s + 1.06i·23-s + 0.645·25-s + 0.305i·29-s + 0.458i·31-s + (1.18 − 0.501i)35-s − 1.46·37-s − 0.546·41-s + 0.0245·43-s + 0.779·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.391 + 0.920i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4806691187\)
\(L(\frac12)\) \(\approx\) \(0.4806691187\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.43 - 1.03i)T \)
good5 \( 1 + 2.86T + 5T^{2} \)
11 \( 1 - 3.32iT - 11T^{2} \)
13 \( 1 - 0.821iT - 13T^{2} \)
17 \( 1 + 3.52T + 17T^{2} \)
19 \( 1 + 5.25iT - 19T^{2} \)
23 \( 1 - 5.12iT - 23T^{2} \)
29 \( 1 - 1.64iT - 29T^{2} \)
31 \( 1 - 2.55iT - 31T^{2} \)
37 \( 1 + 8.93T + 37T^{2} \)
41 \( 1 + 3.49T + 41T^{2} \)
43 \( 1 - 0.161T + 43T^{2} \)
47 \( 1 - 5.34T + 47T^{2} \)
53 \( 1 - 3.28iT - 53T^{2} \)
59 \( 1 + 4.08T + 59T^{2} \)
61 \( 1 + 8.43iT - 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 - 5.68iT - 71T^{2} \)
73 \( 1 + 7.86iT - 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + 12.8iT - 97T^{2} \)
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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

−8.680357327642478314827492530457, −7.74193790119432934720033157727, −6.98060191012731629590044084113, −6.67054657713902391289120145970, −5.38575526933600008083989948927, −4.62223227559352120519340877032, −3.80069908627757813672294082977, −3.05474322434251340963436389186, −1.94693940163061381889103323673, −0.22505942969715729428366499182, 0.73360907680076960474242873837, 2.44057515698065392470785709487, 3.64746420874310365167890981170, 3.76921888091181510867586486769, 4.90177250721548818467434506991, 5.98807126761026179297619916653, 6.61761161110791644830771998142, 7.40521278811407055897614625061, 8.182533645368686374186354452781, 8.650305528389955840911567815693

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