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  + 1.87·2-s + 1.28·3-s + 2.53·4-s + 0.684·5-s + 2.41·6-s − 7-s + 2.87·8-s + 0.652·9-s + 1.28·10-s − 11-s + 3.25·12-s − 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s − 1.96·17-s + 1.22·18-s + 1.73·20-s − 1.28·21-s − 1.87·22-s + 23-s + 3.70·24-s − 0.532·25-s − 3.25·26-s − 0.446·27-s − 2.53·28-s + 1.53·29-s + ⋯
L(s)  = 1  + 1.87·2-s + 1.28·3-s + 2.53·4-s + 0.684·5-s + 2.41·6-s − 7-s + 2.87·8-s + 0.652·9-s + 1.28·10-s − 11-s + 3.25·12-s − 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s − 1.96·17-s + 1.22·18-s + 1.73·20-s − 1.28·21-s − 1.87·22-s + 23-s + 3.70·24-s − 0.532·25-s − 3.25·26-s − 0.446·27-s − 2.53·28-s + 1.53·29-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  =  $(2,\ 3311,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $5.269367784$
$L(\frac12)$  $\approx$  $5.269367784$
$L(1)$   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 \( 1 - 1.87T + T^{2} \)
3 \( 1 - 1.28T + T^{2} \)
5 \( 1 - 0.684T + T^{2} \)
13 \( 1 + 1.73T + T^{2} \)
17 \( 1 + 1.96T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 - 1.53T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.96T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.347T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.87T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.28T + T^{2} \)
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

−8.894989421487216382683199477508, −7.69700929506399408340021405204, −7.17787965206611912902088614782, −6.42735368737720169199084908314, −5.69235973575059459192737165855, −4.77662811109005533339389124303, −4.22750218161341761354355288077, −3.07792495138543732068890134561, −2.47600462797563496725773151650, −2.32042870900223378047411027926, 2.32042870900223378047411027926, 2.47600462797563496725773151650, 3.07792495138543732068890134561, 4.22750218161341761354355288077, 4.77662811109005533339389124303, 5.69235973575059459192737165855, 6.42735368737720169199084908314, 7.17787965206611912902088614782, 7.69700929506399408340021405204, 8.894989421487216382683199477508

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