Properties

Degree 2
Conductor $ 2^{4} \cdot 131 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 13-s + 1.41i·17-s − 1.41i·19-s + 21-s − 1.41i·23-s − 25-s − 27-s + 1.41i·31-s + 1.41i·37-s + 39-s − 41-s + 43-s − 1.41i·47-s + ⋯
L(s)  = 1  + 3-s + 7-s + 13-s + 1.41i·17-s − 1.41i·19-s + 21-s − 1.41i·23-s − 25-s − 27-s + 1.41i·31-s + 1.41i·37-s + 39-s − 41-s + 43-s − 1.41i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2096\)    =    \(2^{4} \cdot 131\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{2096} (785, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2096,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.780551278\)
\(L(\frac12)\)  \(\approx\)  \(1.780551278\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;131\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;131\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
131 \( 1 - T \)
good3 \( 1 - T + T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + 1.41iT - T^{2} \)
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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

−8.906660853001429182504110136387, −8.464597306877219957717274200353, −8.155892555985593367836941257507, −7.04022234099292925304177838587, −6.22539594181083112204551082304, −5.21485591405639881967649004281, −4.29525278779773898392902697755, −3.46469875050591817784900723572, −2.44972261937815343592894593931, −1.50031052699412350748376119261, 1.50502347294976308315535140653, 2.42780047548210021659517563906, 3.54494928420300010174921765875, 4.15949182907551378166290375170, 5.44065126750227160110801092205, 5.94267811948929661101334852475, 7.37468411533620249800865858194, 7.82840153875612647320960060190, 8.405172274389169213269541403539, 9.312736393864224081467563572169

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