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  − 2.21i·5-s + i·7-s + 0.614·11-s − 5.56·13-s + 0.585i·17-s − 3.89i·19-s + 0.850·23-s + 0.0963·25-s + 4.74i·29-s + 3.07i·31-s + 2.21·35-s + 2.17·37-s − 6.07i·41-s − 9.21i·43-s + 3.41·47-s + ⋯
L(s)  = 1  − 0.990i·5-s + 0.377i·7-s + 0.185·11-s − 1.54·13-s + 0.142i·17-s − 0.894i·19-s + 0.177·23-s + 0.0192·25-s + 0.880i·29-s + 0.551i·31-s + 0.374·35-s + 0.357·37-s − 0.949i·41-s − 1.40i·43-s + 0.498·47-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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $0.1802044023$
$L(\frac12)$  $\approx$  $0.1802044023$
$L(\frac{3}{2})$   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 \( 1 + 2.21iT - 5T^{2} \)
11 \( 1 - 0.614T + 11T^{2} \)
13 \( 1 + 5.56T + 13T^{2} \)
17 \( 1 - 0.585iT - 17T^{2} \)
19 \( 1 + 3.89iT - 19T^{2} \)
23 \( 1 - 0.850T + 23T^{2} \)
29 \( 1 - 4.74iT - 29T^{2} \)
31 \( 1 - 3.07iT - 31T^{2} \)
37 \( 1 - 2.17T + 37T^{2} \)
41 \( 1 + 6.07iT - 41T^{2} \)
43 \( 1 + 9.21iT - 43T^{2} \)
47 \( 1 - 3.41T + 47T^{2} \)
53 \( 1 - 4.47iT - 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 + 4.83iT - 67T^{2} \)
71 \( 1 - 5.33T + 71T^{2} \)
73 \( 1 + 9.28T + 73T^{2} \)
79 \( 1 - 1.29iT - 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 - 7.27T + 97T^{2} \)
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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

−8.470697791636649508843040085253, −7.55569672737189024083878294688, −7.06700739464062265850501530398, −6.17854034610057696570148782150, −5.14764288468677140425724604209, −4.98910244715915745001338336407, −4.10348745094942347156076789769, −3.00508279960514779968296908214, −2.21407394102098134448469724813, −1.14398684132876561380804447281, 0.04667426789876708475996127399, 1.53969361647884991701933662989, 2.63405569111159572147835280757, 3.12639352942696345035140030263, 4.22830980979654164006182449302, 4.76354443235916262603031847986, 5.86514481279910528626428603937, 6.38521557257854512805747326656, 7.28240127978392551947883795845, 7.56533314793010342402243215766

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