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  − 0.347·2-s − 0.684·3-s − 0.879·4-s + 1.96·5-s + 0.237·6-s − 7-s + 0.652·8-s − 0.532·9-s − 0.684·10-s − 11-s + 0.601·12-s + 1.73·13-s + 0.347·14-s − 1.34·15-s + 0.652·16-s − 1.28·17-s + 0.184·18-s − 1.73·20-s + 0.684·21-s + 0.347·22-s + 23-s − 0.446·24-s + 2.87·25-s − 0.601·26-s + 1.04·27-s + 0.879·28-s − 1.87·29-s + ⋯
L(s)  = 1  − 0.347·2-s − 0.684·3-s − 0.879·4-s + 1.96·5-s + 0.237·6-s − 7-s + 0.652·8-s − 0.532·9-s − 0.684·10-s − 11-s + 0.601·12-s + 1.73·13-s + 0.347·14-s − 1.34·15-s + 0.652·16-s − 1.28·17-s + 0.184·18-s − 1.73·20-s + 0.684·21-s + 0.347·22-s + 23-s − 0.446·24-s + 2.87·25-s − 0.601·26-s + 1.04·27-s + 0.879·28-s − 1.87·29-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  =  $(2,\ 3311,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.7893488418$
$L(\frac12)$  $\approx$  $0.7893488418$
$L(1)$   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 \( 1 + 0.347T + T^{2} \)
3 \( 1 + 0.684T + T^{2} \)
5 \( 1 - 1.96T + T^{2} \)
13 \( 1 - 1.73T + T^{2} \)
17 \( 1 + 1.28T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + 1.87T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.28T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.53T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 0.347T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.684T + T^{2} \)
89 \( 1 + T^{2} \)
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

−9.114437117079700269265617770829, −8.440380645133613988518787430808, −7.13069396333858338299453190414, −6.32718592716134258673214753593, −5.62474712373321787914037586107, −5.49650302070396175891813836480, −4.30592251167204352831845041925, −3.12834159359762304708597431312, −2.16954680516121578815263408390, −0.864784803723543020446377103744, 0.864784803723543020446377103744, 2.16954680516121578815263408390, 3.12834159359762304708597431312, 4.30592251167204352831845041925, 5.49650302070396175891813836480, 5.62474712373321787914037586107, 6.32718592716134258673214753593, 7.13069396333858338299453190414, 8.440380645133613988518787430808, 9.114437117079700269265617770829

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