Properties

Degree 2
Conductor $ 11 \cdot 2797 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 2·4-s − 2·5-s + 6·6-s − 3·7-s + 6·9-s + 4·10-s + 11-s − 6·12-s − 6·13-s + 6·14-s + 6·15-s − 4·16-s − 5·17-s − 12·18-s − 4·19-s − 4·20-s + 9·21-s − 2·22-s − 2·23-s − 25-s + 12·26-s − 9·27-s − 6·28-s − 9·29-s − 12·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s − 1.13·7-s + 2·9-s + 1.26·10-s + 0.301·11-s − 1.73·12-s − 1.66·13-s + 1.60·14-s + 1.54·15-s − 16-s − 1.21·17-s − 2.82·18-s − 0.917·19-s − 0.894·20-s + 1.96·21-s − 0.426·22-s − 0.417·23-s − 1/5·25-s + 2.35·26-s − 1.73·27-s − 1.13·28-s − 1.67·29-s − 2.19·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 30767 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 30767 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30767\)    =    \(11 \cdot 2797\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30767} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 30767,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{11,\;2797\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{11,\;2797\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 - T \)
2797 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.12122894819405, −15.76628821869838, −15.12038987731355, −14.65682879174419, −13.49285931717956, −13.01173139776716, −12.41135631108765, −12.12262138681777, −11.37399396487313, −11.08689489675127, −10.58956558889926, −9.929743311140841, −9.617988943679059, −9.022661700255221, −8.361929904272109, −7.440125420687716, −7.196941034939674, −6.754242530831624, −6.163184401108157, −5.383987503007182, −4.729809645262169, −4.134770771763613, −3.445644736809221, −2.146763543495326, −1.621254120426992, 0, 0, 0, 1.621254120426992, 2.146763543495326, 3.445644736809221, 4.134770771763613, 4.729809645262169, 5.383987503007182, 6.163184401108157, 6.754242530831624, 7.196941034939674, 7.440125420687716, 8.361929904272109, 9.022661700255221, 9.617988943679059, 9.929743311140841, 10.58956558889926, 11.08689489675127, 11.37399396487313, 12.12262138681777, 12.41135631108765, 13.01173139776716, 13.49285931717956, 14.65682879174419, 15.12038987731355, 15.76628821869838, 16.12122894819405

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