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 + 3.30·3-s + 4-s − 5-s + 3.30·6-s − 0.676·7-s + 8-s + 7.92·9-s − 10-s + 11-s + 3.30·12-s − 3.96·13-s − 0.676·14-s − 3.30·15-s + 16-s − 2.43·17-s + 7.92·18-s + 1.47·19-s − 20-s − 2.23·21-s + 22-s + 3.38·23-s + 3.30·24-s + 25-s − 3.96·26-s + 16.2·27-s − 0.676·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.90·3-s + 0.5·4-s − 0.447·5-s + 1.34·6-s − 0.255·7-s + 0.353·8-s + 2.64·9-s − 0.316·10-s + 0.301·11-s + 0.953·12-s − 1.09·13-s − 0.180·14-s − 0.853·15-s + 0.250·16-s − 0.591·17-s + 1.86·18-s + 0.339·19-s − 0.223·20-s − 0.488·21-s + 0.213·22-s + 0.706·23-s + 0.674·24-s + 0.200·25-s − 0.776·26-s + 3.12·27-s − 0.127·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.032380048$
$L(\frac12)$  $\approx$  $6.032380048$
$L(\frac{3}{2})$   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 \( 1 - 3.30T + 3T^{2} \)
7 \( 1 + 0.676T + 7T^{2} \)
13 \( 1 + 3.96T + 13T^{2} \)
17 \( 1 + 2.43T + 17T^{2} \)
19 \( 1 - 1.47T + 19T^{2} \)
23 \( 1 - 3.38T + 23T^{2} \)
29 \( 1 - 2.44T + 29T^{2} \)
31 \( 1 - 8.31T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 + 4.31T + 41T^{2} \)
47 \( 1 - 7.27T + 47T^{2} \)
53 \( 1 - 9.23T + 53T^{2} \)
59 \( 1 - 1.87T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + 2.45T + 67T^{2} \)
71 \( 1 - 8.83T + 71T^{2} \)
73 \( 1 - 5.06T + 73T^{2} \)
79 \( 1 - 6.51T + 79T^{2} \)
83 \( 1 + 5.30T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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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

−8.206189781743159911217882507764, −7.60327832528600250596764491394, −7.01629011439945047187164217012, −6.35104416302650406813602772577, −4.95106423123996186009142879215, −4.37840281062980658901132074869, −3.71338959585507372358298711356, −2.74034900197146233052287111786, −2.54252782399914712132463565058, −1.20984014167860996981471399697, 1.20984014167860996981471399697, 2.54252782399914712132463565058, 2.74034900197146233052287111786, 3.71338959585507372358298711356, 4.37840281062980658901132074869, 4.95106423123996186009142879215, 6.35104416302650406813602772577, 7.01629011439945047187164217012, 7.60327832528600250596764491394, 8.206189781743159911217882507764

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