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 − 2.61·3-s + 4-s − 5-s − 2.61·6-s + 0.881·7-s + 8-s + 3.83·9-s − 10-s + 11-s − 2.61·12-s − 6.07·13-s + 0.881·14-s + 2.61·15-s + 16-s − 6.61·17-s + 3.83·18-s − 1.59·19-s − 20-s − 2.30·21-s + 22-s + 1.90·23-s − 2.61·24-s + 25-s − 6.07·26-s − 2.17·27-s + 0.881·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.50·3-s + 0.5·4-s − 0.447·5-s − 1.06·6-s + 0.333·7-s + 0.353·8-s + 1.27·9-s − 0.316·10-s + 0.301·11-s − 0.754·12-s − 1.68·13-s + 0.235·14-s + 0.674·15-s + 0.250·16-s − 1.60·17-s + 0.902·18-s − 0.365·19-s − 0.223·20-s − 0.502·21-s + 0.213·22-s + 0.396·23-s − 0.533·24-s + 0.200·25-s − 1.19·26-s − 0.417·27-s + 0.166·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.101666507$
$L(\frac12)$  $\approx$  $1.101666507$
$L(\frac{3}{2})$   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 \( 1 + 2.61T + 3T^{2} \)
7 \( 1 - 0.881T + 7T^{2} \)
13 \( 1 + 6.07T + 13T^{2} \)
17 \( 1 + 6.61T + 17T^{2} \)
19 \( 1 + 1.59T + 19T^{2} \)
23 \( 1 - 1.90T + 23T^{2} \)
29 \( 1 + 3.16T + 29T^{2} \)
31 \( 1 - 7.78T + 31T^{2} \)
37 \( 1 - 3.74T + 37T^{2} \)
41 \( 1 - 7.16T + 41T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 - 8.84T + 53T^{2} \)
59 \( 1 - 4.82T + 59T^{2} \)
61 \( 1 + 6.29T + 61T^{2} \)
67 \( 1 + 7.78T + 67T^{2} \)
71 \( 1 - 4.25T + 71T^{2} \)
73 \( 1 + 6.44T + 73T^{2} \)
79 \( 1 - 5.91T + 79T^{2} \)
83 \( 1 + 8.39T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 - 3.95T + 97T^{2} \)
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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

−8.040832162709309838415101420880, −7.25296715347834276897990943578, −6.67584651436143416303622955904, −6.11390319167596728208911573398, −5.20907856932957070487598729645, −4.57957249681750582113364545279, −4.32160959793454215085899639009, −2.90777095528726503020850869668, −1.95305391619183217558004266197, −0.54476433623510230454075147764, 0.54476433623510230454075147764, 1.95305391619183217558004266197, 2.90777095528726503020850869668, 4.32160959793454215085899639009, 4.57957249681750582113364545279, 5.20907856932957070487598729645, 6.11390319167596728208911573398, 6.67584651436143416303622955904, 7.25296715347834276897990943578, 8.040832162709309838415101420880

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