Properties

Degree 2
Conductor $ 2^{8} \cdot 3^{2} $
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  + 2i·5-s − 4·7-s − 4i·11-s − 2i·13-s + 6·17-s + 4i·19-s + 25-s − 2i·29-s − 4·31-s − 8i·35-s + 2i·37-s + 2·41-s + 4i·43-s + 8·47-s + 9·49-s + ⋯
L(s)  = 1  + 0.894i·5-s − 1.51·7-s − 1.20i·11-s − 0.554i·13-s + 1.45·17-s + 0.917i·19-s + 0.200·25-s − 0.371i·29-s − 0.718·31-s − 1.35i·35-s + 0.328i·37-s + 0.312·41-s + 0.609i·43-s + 1.16·47-s + 1.28·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2304\)    =    \(2^{8} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.707 - 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{2304} (1153, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2304,\ (\ :1/2),\ 0.707 - 0.707i)\)
\(L(1)\)  \(\approx\)  \(1.352639401\)
\(L(\frac12)\)  \(\approx\)  \(1.352639401\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−9.261463209821477941125778635800, −8.236843162783280849131552804236, −7.54098490973968202557563080677, −6.69180965880791105959217480226, −5.95904907682833684919655114106, −5.53859079119232906435950130673, −3.87076273697347354838757659673, −3.26844259369493471845001666673, −2.70317677274366157697142455996, −0.888605342906179785623422702376, 0.62769668396498882839898116695, 1.99818597084615958384961295381, 3.17169361119004245545314740202, 4.02882779336633236721320512629, 4.97782031131378739858302163881, 5.67422571615892971359132728138, 6.77209989029074272953434809146, 7.16715810554002748224133427983, 8.206271792843165666601521568412, 9.196157733225529089312465358172

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