Properties

Degree 2
Conductor $ 2^{3} \cdot 227 $
Sign $-i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·5-s + 7-s − 9-s − 11-s + 1.41i·17-s + 19-s + 23-s − 1.00·25-s − 29-s + 1.41i·31-s + 1.41i·35-s − 1.41i·37-s + 1.41i·41-s − 43-s − 1.41i·45-s + ⋯
L(s)  = 1  + 1.41i·5-s + 7-s − 9-s − 11-s + 1.41i·17-s + 19-s + 23-s − 1.00·25-s − 29-s + 1.41i·31-s + 1.41i·35-s − 1.41i·37-s + 1.41i·41-s − 43-s − 1.41i·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1816\)    =    \(2^{3} \cdot 227\)
\( \varepsilon \)  =  $-i$
motivic weight  =  \(0\)
character  :  $\chi_{1816} (1361, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1816,\ (\ :0),\ -i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.051521405\)
\(L(\frac12)\)  \(\approx\)  \(1.051521405\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;227\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;227\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
227 \( 1 - T \)
good3 \( 1 + T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.782869674073882682294966406179, −8.690591696004891243407073384139, −8.014953661191686189037879366085, −7.38105736426411028548569336809, −6.50022245402056320541513474539, −5.59908845275094230217614948124, −4.94962662799138920379017218321, −3.52155864407868966022147791962, −2.89757931314720110339563724093, −1.81636201304736489821123886462, 0.793535492475786491178870444877, 2.16497633502875272725977626500, 3.27044158298958439696083253953, 4.67603848224771782950109597715, 5.18919013112210496284495211707, 5.59433610673138138826891755980, 7.07975163521363885129358929140, 7.965054212469885151445703335301, 8.357373828857800543510227085131, 9.229819229781344398938275653051

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