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 − 2.31·3-s + 4-s + 5-s + 2.31·6-s + 4.25·7-s − 8-s + 2.37·9-s − 10-s + 11-s − 2.31·12-s + 2.14·13-s − 4.25·14-s − 2.31·15-s + 16-s + 0.399·17-s − 2.37·18-s − 0.398·19-s + 20-s − 9.85·21-s − 22-s − 1.33·23-s + 2.31·24-s + 25-s − 2.14·26-s + 1.45·27-s + 4.25·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.33·3-s + 0.5·4-s + 0.447·5-s + 0.946·6-s + 1.60·7-s − 0.353·8-s + 0.790·9-s − 0.316·10-s + 0.301·11-s − 0.669·12-s + 0.595·13-s − 1.13·14-s − 0.598·15-s + 0.250·16-s + 0.0969·17-s − 0.559·18-s − 0.0914·19-s + 0.223·20-s − 2.15·21-s − 0.213·22-s − 0.277·23-s + 0.473·24-s + 0.200·25-s − 0.421·26-s + 0.279·27-s + 0.803·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.249676341$
$L(\frac12)$  $\approx$  $1.249676341$
$L(\frac{3}{2})$   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 \( 1 + 2.31T + 3T^{2} \)
7 \( 1 - 4.25T + 7T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
17 \( 1 - 0.399T + 17T^{2} \)
19 \( 1 + 0.398T + 19T^{2} \)
23 \( 1 + 1.33T + 23T^{2} \)
29 \( 1 + 4.75T + 29T^{2} \)
31 \( 1 - 7.39T + 31T^{2} \)
37 \( 1 - 0.118T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
47 \( 1 + 3.10T + 47T^{2} \)
53 \( 1 - 6.55T + 53T^{2} \)
59 \( 1 + 3.06T + 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 + 5.14T + 73T^{2} \)
79 \( 1 + 5.53T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + 5.68T + 97T^{2} \)
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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

−8.313126129063096719554358835432, −7.60741525981249303107328923608, −6.81236843236228472875390703003, −6.02521262109244354278884204777, −5.55859856583602238251007711278, −4.77954174078229624376257404093, −4.00918509716241890541387902632, −2.51003255113544496841817516575, −1.52640497433050082709337696877, −0.811668808656325173656208869953, 0.811668808656325173656208869953, 1.52640497433050082709337696877, 2.51003255113544496841817516575, 4.00918509716241890541387902632, 4.77954174078229624376257404093, 5.55859856583602238251007711278, 6.02521262109244354278884204777, 6.81236843236228472875390703003, 7.60741525981249303107328923608, 8.313126129063096719554358835432

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