Properties

Degree 2
Conductor $ 7 \cdot 11 \cdot 43 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s + 11-s + 2·13-s − 14-s + 15-s − 16-s − 17-s − 21-s − 22-s + 2·23-s − 24-s − 2·26-s + 27-s − 29-s − 30-s − 33-s + 34-s − 35-s − 2·39-s − 40-s − 41-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 7-s + 8-s + 10-s + 11-s + 2·13-s − 14-s + 15-s − 16-s − 17-s − 21-s − 22-s + 2·23-s − 24-s − 2·26-s + 27-s − 29-s − 30-s − 33-s + 34-s − 35-s − 2·39-s − 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3311\)    =    \(7 \cdot 11 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{3311} (3310, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3311,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.4789155944$
$L(\frac12)$  $\approx$  $0.4789155944$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;11,\;43\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{7,\;11,\;43\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 - T \)
11 \( 1 - T \)
43 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 + T + T^{2} \)
5 \( 1 + T + T^{2} \)
13 \( ( 1 - T )^{2} \)
17 \( 1 + T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )^{2} \)
29 \( 1 + T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T + T^{2} \)
89 \( ( 1 - T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.862262329947913428343529070066, −8.258538725477221090726449746241, −7.43927517609107944454842365043, −6.72363180263083440484828215114, −5.90167615767299762479450629027, −4.90290354737603484033044377537, −4.29092155912826760244323363959, −3.47490674582878888548143372136, −1.63069526623862256761644080483, −0.819001488182938398376713417081, 0.819001488182938398376713417081, 1.63069526623862256761644080483, 3.47490674582878888548143372136, 4.29092155912826760244323363959, 4.90290354737603484033044377537, 5.90167615767299762479450629027, 6.72363180263083440484828215114, 7.43927517609107944454842365043, 8.258538725477221090726449746241, 8.862262329947913428343529070066

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