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  + 2·2-s + 3·4-s + 7-s + 4·8-s − 9-s − 11-s + 2·14-s + 5·16-s − 2·18-s − 2·22-s − 2·23-s − 25-s + 3·28-s − 2·29-s + 6·32-s − 3·36-s − 43-s − 3·44-s − 4·46-s + 49-s − 2·50-s + 2·53-s + 4·56-s − 4·58-s − 63-s + 7·64-s + 2·67-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 7-s + 4·8-s − 9-s − 11-s + 2·14-s + 5·16-s − 2·18-s − 2·22-s − 2·23-s − 25-s + 3·28-s − 2·29-s + 6·32-s − 3·36-s − 43-s − 3·44-s − 4·46-s + 49-s − 2·50-s + 2·53-s + 4·56-s − 4·58-s − 63-s + 7·64-s + 2·67-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  =  $(2,\ 3311,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $4.436512507$
$L(\frac12)$  $\approx$  $4.436512507$
$L(1)$   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 \( ( 1 - T )^{2} \)
3 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 + T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
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

−8.327529048966035595447667808121, −7.86576770894213139238402564395, −7.21276800265659617620724937401, −6.10546057474397191428028034742, −5.55683797397808236356124878961, −5.16058185068508082676236724432, −4.12271596725253729989644227068, −3.56122613556173330407350824039, −2.37195125549253654885749496071, −1.94810746962855116347226891179, 1.94810746962855116347226891179, 2.37195125549253654885749496071, 3.56122613556173330407350824039, 4.12271596725253729989644227068, 5.16058185068508082676236724432, 5.55683797397808236356124878961, 6.10546057474397191428028034742, 7.21276800265659617620724937401, 7.86576770894213139238402564395, 8.327529048966035595447667808121

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