Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.525 + 0.850i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.69·5-s i·7-s + 2.05i·11-s − 4.65i·13-s − 5.73i·17-s − 3.27·19-s − 4.45·23-s + 8.66·25-s − 10.2·29-s − 1.26i·31-s − 3.69i·35-s − 3.25i·37-s − 8.72i·41-s + 2.56·43-s + 2.03·47-s + ⋯
L(s)  = 1  + 1.65·5-s − 0.377i·7-s + 0.620i·11-s − 1.28i·13-s − 1.39i·17-s − 0.750·19-s − 0.928·23-s + 1.73·25-s − 1.90·29-s − 0.226i·31-s − 0.624i·35-s − 0.535i·37-s − 1.36i·41-s + 0.391·43-s + 0.296·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.525 + 0.850i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.525 + 0.850i)$
$L(1)$  $\approx$  $1.736408042$
$L(\frac12)$  $\approx$  $1.736408042$
$L(\frac{3}{2})$   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 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 - 2.05iT - 11T^{2} \)
13 \( 1 + 4.65iT - 13T^{2} \)
17 \( 1 + 5.73iT - 17T^{2} \)
19 \( 1 + 3.27T + 19T^{2} \)
23 \( 1 + 4.45T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
37 \( 1 + 3.25iT - 37T^{2} \)
41 \( 1 + 8.72iT - 41T^{2} \)
43 \( 1 - 2.56T + 43T^{2} \)
47 \( 1 - 2.03T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 + 0.851iT - 59T^{2} \)
61 \( 1 - 2.31iT - 61T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 + 4.44T + 71T^{2} \)
73 \( 1 + 6.97T + 73T^{2} \)
79 \( 1 + 11.6iT - 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + 2.35iT - 89T^{2} \)
97 \( 1 - 4.02T + 97T^{2} \)
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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

−7.55831475513020175479268120893, −7.29720600003949948347097645199, −6.20920378226991828367751603715, −5.74523443283049524390609126772, −5.12856150138385868370571919612, −4.27638833531119017735434879275, −3.20161974334436224383202072858, −2.32118944200986004464465115032, −1.70146802340549629309170916432, −0.37184361073096401417524443320, 1.66874583754443467503229268904, 1.84347224376130460011251938659, 2.93108526535656183772164849187, 3.96829869305781368091254157386, 4.74775683096026744323799738730, 5.80529638527368780024974000637, 6.05790982591678072211871740110, 6.58111441943891266733749751350, 7.62850963065032893717969677709, 8.523517470153322018987860091770

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