Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.72·3-s + 4-s + 5-s + 2.72·6-s + 1.91·7-s + 8-s + 4.40·9-s + 10-s − 11-s + 2.72·12-s + 1.20·13-s + 1.91·14-s + 2.72·15-s + 16-s − 3.77·17-s + 4.40·18-s + 5.93·19-s + 20-s + 5.20·21-s − 22-s + 0.791·23-s + 2.72·24-s + 25-s + 1.20·26-s + 3.81·27-s + 1.91·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.57·3-s + 0.5·4-s + 0.447·5-s + 1.11·6-s + 0.722·7-s + 0.353·8-s + 1.46·9-s + 0.316·10-s − 0.301·11-s + 0.785·12-s + 0.335·13-s + 0.511·14-s + 0.702·15-s + 0.250·16-s − 0.915·17-s + 1.03·18-s + 1.36·19-s + 0.223·20-s + 1.13·21-s − 0.213·22-s + 0.164·23-s + 0.555·24-s + 0.200·25-s + 0.237·26-s + 0.733·27-s + 0.361·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4730\)    =    \(2 \cdot 5 \cdot 11 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4730} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.740600273$
$L(\frac12)$  $\approx$  $6.740600273$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 + T \)
43 \( 1 - T \)
good3 \( 1 - 2.72T + 3T^{2} \)
7 \( 1 - 1.91T + 7T^{2} \)
13 \( 1 - 1.20T + 13T^{2} \)
17 \( 1 + 3.77T + 17T^{2} \)
19 \( 1 - 5.93T + 19T^{2} \)
23 \( 1 - 0.791T + 23T^{2} \)
29 \( 1 - 0.597T + 29T^{2} \)
31 \( 1 - 7.15T + 31T^{2} \)
37 \( 1 - 1.12T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 + 4.42T + 59T^{2} \)
61 \( 1 + 2.65T + 61T^{2} \)
67 \( 1 - 15.5T + 67T^{2} \)
71 \( 1 + 4.84T + 71T^{2} \)
73 \( 1 + 2.10T + 73T^{2} \)
79 \( 1 + 0.235T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 1.84T + 89T^{2} \)
97 \( 1 + 0.701T + 97T^{2} \)
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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

−8.203360490234534137749988371210, −7.75589762413770962423476998051, −6.88202911648394701220315095649, −6.17834976487785439265138726487, −5.01937497258222161683570802693, −4.65197645410955750567020755122, −3.50918105387691102357720283281, −3.01108214266681097424486074054, −2.12716079660360227964765324649, −1.40518332240939556255518970719, 1.40518332240939556255518970719, 2.12716079660360227964765324649, 3.01108214266681097424486074054, 3.50918105387691102357720283281, 4.65197645410955750567020755122, 5.01937497258222161683570802693, 6.17834976487785439265138726487, 6.88202911648394701220315095649, 7.75589762413770962423476998051, 8.203360490234534137749988371210

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