Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.707 - 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.896i·5-s i·7-s + 3.34·11-s + 4·13-s + 3.48i·17-s + i·19-s + 0.138·23-s + 4.19·25-s + 2.07i·29-s + 0.267i·31-s + 0.896·35-s − 8.26·37-s + 9.28i·41-s + 0.535i·43-s + 5.93·47-s + ⋯
L(s)  = 1  + 0.400i·5-s − 0.377i·7-s + 1.00·11-s + 1.10·13-s + 0.845i·17-s + 0.229i·19-s + 0.0289·23-s + 0.839·25-s + 0.384i·29-s + 0.0481i·31-s + 0.151·35-s − 1.35·37-s + 1.44i·41-s + 0.0817i·43-s + 0.865·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.707 - 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $2.255592247$
$L(\frac12)$  $\approx$  $2.255592247$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good5 \( 1 - 0.896iT - 5T^{2} \)
11 \( 1 - 3.34T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 3.48iT - 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 - 0.138T + 23T^{2} \)
29 \( 1 - 2.07iT - 29T^{2} \)
31 \( 1 - 0.267iT - 31T^{2} \)
37 \( 1 + 8.26T + 37T^{2} \)
41 \( 1 - 9.28iT - 41T^{2} \)
43 \( 1 - 0.535iT - 43T^{2} \)
47 \( 1 - 5.93T + 47T^{2} \)
53 \( 1 + 2.82iT - 53T^{2} \)
59 \( 1 + 1.03T + 59T^{2} \)
61 \( 1 - 8.39T + 61T^{2} \)
67 \( 1 + 2.92iT - 67T^{2} \)
71 \( 1 - 1.55T + 71T^{2} \)
73 \( 1 - 6.39T + 73T^{2} \)
79 \( 1 + 0.535iT - 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + 10.7iT - 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−8.341614423302238742825226659303, −7.33408627950546573042265696578, −6.66951953799077369525195877714, −6.22626451062727744807128599723, −5.36994280828681837047760416270, −4.36903787924627294122337912507, −3.71130790043618205693535224500, −3.09763536587661317739924617664, −1.80167273418313417034843478182, −1.03671576512556810368373951686, 0.69069654014593368674084607377, 1.61883407155371568127029743921, 2.67713297597380962947425978184, 3.63428451419099202405294900562, 4.26275848952264622575817298516, 5.22026063140596292140902650626, 5.75607935970639950548268537722, 6.69776360129655908820600740525, 7.08867130346602775947286838708, 8.155571843365050006472495970323

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