Properties

Degree 2
Conductor $ 2 \cdot 5821 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 4-s − 3·5-s + 3·6-s − 4·7-s − 8-s + 6·9-s + 3·10-s − 6·11-s − 3·12-s − 6·13-s + 4·14-s + 9·15-s + 16-s − 6·17-s − 6·18-s − 5·19-s − 3·20-s + 12·21-s + 6·22-s − 23-s + 3·24-s + 4·25-s + 6·26-s − 9·27-s − 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s + 1.22·6-s − 1.51·7-s − 0.353·8-s + 2·9-s + 0.948·10-s − 1.80·11-s − 0.866·12-s − 1.66·13-s + 1.06·14-s + 2.32·15-s + 1/4·16-s − 1.45·17-s − 1.41·18-s − 1.14·19-s − 0.670·20-s + 2.61·21-s + 1.27·22-s − 0.208·23-s + 0.612·24-s + 4/5·25-s + 1.17·26-s − 1.73·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11642 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11642 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11642\)    =    \(2 \cdot 5821\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{11642} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 11642,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5821\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5821\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5821 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−17.22470689892625, −16.67465320618330, −16.16735340282855, −15.83709879028198, −15.23087896041019, −15.07040410816940, −13.50636739785262, −12.93519174449010, −12.34135699077555, −12.30610968238024, −11.41541318641353, −10.75782779681942, −10.56701518079405, −9.966018390070901, −9.207629260574627, −8.475300577436567, −7.495091070379581, −7.161477709691316, −6.807044431585419, −5.844011817489637, −5.350212081651227, −4.545192854519453, −3.890479016237367, −2.845222351133270, −2.031837058613338, 0, 0, 0, 2.031837058613338, 2.845222351133270, 3.890479016237367, 4.545192854519453, 5.350212081651227, 5.844011817489637, 6.807044431585419, 7.161477709691316, 7.495091070379581, 8.475300577436567, 9.207629260574627, 9.966018390070901, 10.56701518079405, 10.75782779681942, 11.41541318641353, 12.30610968238024, 12.34135699077555, 12.93519174449010, 13.50636739785262, 15.07040410816940, 15.23087896041019, 15.83709879028198, 16.16735340282855, 16.67465320618330, 17.22470689892625

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