Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23·2-s − 3-s + 3.00·4-s + 1.61·5-s − 2.23·6-s − 7-s + 2.23·8-s + 9-s + 3.61·10-s − 3.00·12-s + 1.23·13-s − 2.23·14-s − 1.61·15-s − 0.999·16-s + 1.85·17-s + 2.23·18-s + 3.85·19-s + 4.85·20-s + 21-s + 6.61·23-s − 2.23·24-s − 2.38·25-s + 2.76·26-s − 27-s − 3.00·28-s + 6·29-s − 3.61·30-s + ⋯
L(s)  = 1  + 1.58·2-s − 0.577·3-s + 1.50·4-s + 0.723·5-s − 0.912·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s + 1.14·10-s − 0.866·12-s + 0.342·13-s − 0.597·14-s − 0.417·15-s − 0.249·16-s + 0.449·17-s + 0.527·18-s + 0.884·19-s + 1.08·20-s + 0.218·21-s + 1.37·23-s − 0.456·24-s − 0.476·25-s + 0.542·26-s − 0.192·27-s − 0.566·28-s + 1.11·29-s − 0.660·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2541} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2541,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.313229114$
$L(\frac12)$  $\approx$  $4.313229114$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 2.23T + 2T^{2} \)
5 \( 1 - 1.61T + 5T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 - 1.85T + 17T^{2} \)
19 \( 1 - 3.85T + 19T^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8.09T + 31T^{2} \)
37 \( 1 - 2.38T + 37T^{2} \)
41 \( 1 + 9.61T + 41T^{2} \)
43 \( 1 - 3.70T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 1.23T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 0.291T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 7.52T + 83T^{2} \)
89 \( 1 - 3.38T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.023743568221924504216409428378, −7.914415105534707878010448230928, −6.72716092899351277528219788226, −6.51921128365278261715880817369, −5.50879013082944547998349927684, −5.18154083423738583247014252208, −4.23853109921884502856414249422, −3.28835259961409353657930448881, −2.53682875350577112771699235860, −1.15114229669994717867736904528, 1.15114229669994717867736904528, 2.53682875350577112771699235860, 3.28835259961409353657930448881, 4.23853109921884502856414249422, 5.18154083423738583247014252208, 5.50879013082944547998349927684, 6.51921128365278261715880817369, 6.72716092899351277528219788226, 7.914415105534707878010448230928, 9.023743568221924504216409428378

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