Properties

Degree 2
Conductor $ 2^{8} \cdot 3 \cdot 7 $
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 + 7-s − 9-s + 6i·11-s − 2i·13-s − 4i·19-s + i·21-s − 6·23-s + 5·25-s i·27-s − 6i·29-s − 8·31-s − 6·33-s + 2i·37-s + 2·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.377·7-s − 0.333·9-s + 1.80i·11-s − 0.554i·13-s − 0.917i·19-s + 0.218i·21-s − 1.25·23-s + 25-s − 0.192i·27-s − 1.11i·29-s − 1.43·31-s − 1.04·33-s + 0.328i·37-s + 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5376\)    =    \(2^{8} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.707 + 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{5376} (2689, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 5376,\ (\ :1/2),\ -0.707 + 0.707i)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

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

−7.983081919844388091019770191939, −7.16871189149650957205261704801, −6.60145928989985901456930619813, −5.54229606706781697400865913072, −4.87548881225291001669457992762, −4.37511585051108608285918087555, −3.46445598051237037944234230449, −2.44472394532890852593298668173, −1.64127100441604991309627763858, 0, 1.29227534929850042669879402250, 2.06875207797482378266165752435, 3.32011788639021068020984106117, 3.72163334974701637072978281481, 5.02742272292131235884624338243, 5.57844285564960434267849504143, 6.39204791689497160412699890303, 6.89595211369618080201968592524, 7.942696774340037832322275123907

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