Properties

Degree 2
Conductor $ 2^{8} \cdot 3 $
Sign $-0.707 + 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 2i·5-s − 4·7-s − 9-s − 4i·11-s + 2i·13-s − 2·15-s − 6·17-s − 4i·19-s − 4i·21-s + 25-s i·27-s − 2i·29-s − 4·31-s + 4·33-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.894i·5-s − 1.51·7-s − 0.333·9-s − 1.20i·11-s + 0.554i·13-s − 0.516·15-s − 1.45·17-s − 0.917i·19-s − 0.872i·21-s + 0.200·25-s − 0.192i·27-s − 0.371i·29-s − 0.718·31-s + 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(768\)    =    \(2^{8} \cdot 3\)
\( \varepsilon \)  =  $-0.707 + 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{768} (385, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 768,\ (\ :1/2),\ -0.707 + 0.707i)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 - iT \)
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

−9.992657737159952110923654674533, −9.199150600618356363333291423906, −8.605948567033838895002997751855, −7.06992965408008235742663108340, −6.55420735276939294351881900748, −5.73167023113259948249081162165, −4.32137562834720234205738904062, −3.33072250231337258947255629984, −2.61280563257824176573263690653, 0, 1.73207908456882757664741684628, 3.06310481619601238631488333900, 4.29370400578789576461250993928, 5.34799277350386073726216311847, 6.43427622228386769543730958206, 7.02636487220597968841948174938, 8.109292621001179927364041734470, 8.998327639266284776293447036244, 9.687819352404895063426047923888

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