Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.377 - 0.925i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·5-s + (1 − 2.44i)7-s + 4.24i·11-s − 2.44i·13-s − 1.73·17-s − 2.44i·19-s + 8.48i·23-s − 2.00·25-s − 4.24i·29-s + 7.34i·31-s + (−1.73 + 4.24i)35-s + 37-s + 1.73·41-s − 7·43-s + 12.1·47-s + ⋯
L(s)  = 1  − 0.774·5-s + (0.377 − 0.925i)7-s + 1.27i·11-s − 0.679i·13-s − 0.420·17-s − 0.561i·19-s + 1.76i·23-s − 0.400·25-s − 0.787i·29-s + 1.31i·31-s + (−0.292 + 0.717i)35-s + 0.164·37-s + 0.270·41-s − 1.06·43-s + 1.76·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.377 - 0.925i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.377 - 0.925i)\)
\(L(1)\)  \(\approx\)  \(1.146748767\)
\(L(\frac12)\)  \(\approx\)  \(1.146748767\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1 + 2.44i)T \)
good5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 + 2.44iT - 19T^{2} \)
23 \( 1 - 8.48iT - 23T^{2} \)
29 \( 1 + 4.24iT - 29T^{2} \)
31 \( 1 - 7.34iT - 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 1.73T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 8.66T + 59T^{2} \)
61 \( 1 - 2.44iT - 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 2.44iT - 97T^{2} \)
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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

−8.720862094712660525750598068456, −8.003232089585694378602081003928, −7.16875633179951565472967993363, −7.11550294238860296458556176016, −5.70893574452764123888537827355, −4.86566877578844198242221059250, −4.15015694087300773238637750350, −3.47333657536795122779936744238, −2.21938350226781555797689071923, −1.03221639396362627318288334790, 0.41978416099709583996406668142, 1.95610039643426885534331612010, 2.89279373750426527116407326326, 3.89797676505784121590230570039, 4.58848325592802861323520408796, 5.63393415860003010802492175616, 6.21208742997116623237146350222, 7.09610692152086848787401380155, 8.086161472866834679151485858188, 8.502472497445711177332229944698

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