Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.580 + 0.814i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)5-s + (−2 − 1.73i)7-s + (1.5 + 2.59i)11-s + (−0.5 − 0.866i)13-s + (1.5 − 2.59i)17-s + (2.5 + 4.33i)19-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-s + (4.5 − 7.79i)29-s − 4·31-s + (−2.5 + 0.866i)35-s + (−2.5 − 4.33i)37-s + (3.5 + 6.06i)41-s + (1.5 − 2.59i)43-s + 8·47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.755 − 0.654i)7-s + (0.452 + 0.783i)11-s + (−0.138 − 0.240i)13-s + (0.363 − 0.630i)17-s + (0.573 + 0.993i)19-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-s + (0.835 − 1.44i)29-s − 0.718·31-s + (−0.422 + 0.146i)35-s + (−0.410 − 0.711i)37-s + (0.546 + 0.946i)41-s + (0.228 − 0.396i)43-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.580 + 0.814i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (2881, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.580 + 0.814i)\)
\(L(1)\)  \(\approx\)  \(1.708529099\)
\(L(\frac12)\)  \(\approx\)  \(1.708529099\)
\(L(\frac{3}{2})\)   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 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.5 - 6.06i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.5 + 2.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-6.5 + 11.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.732933934379693464505594422958, −7.57119411049803624747703675143, −7.33180214280573488193473806053, −6.29528527917720156835495088659, −5.61272125178578237686975088797, −4.67411412605140816136168538629, −3.87324837390155450402425278720, −3.03455879373448703715894101996, −1.80732380135353407547154501953, −0.65696023828015177518718461867, 1.01104755304490151672534901495, 2.44675504335956544698178453999, 3.11240433503660135053255715929, 3.98919237570411684373141638999, 5.14614998307676859804294781221, 5.85777717362417224918877599729, 6.59503516280162287222229234414, 7.11633652885132385851304770990, 8.233174962578292023469683553017, 8.985913729479718967831380398549

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