Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s − 3-s + 0.618·4-s − 2.23·5-s + 1.61·6-s − 7-s + 2.23·8-s + 9-s + 3.61·10-s − 0.618·12-s − 0.236·13-s + 1.61·14-s + 2.23·15-s − 4.85·16-s − 2·17-s − 1.61·18-s + 1.47·19-s − 1.38·20-s + 21-s − 2·23-s − 2.23·24-s + 0.381·26-s − 27-s − 0.618·28-s − 5·29-s − 3.61·30-s + 10.4·31-s + ⋯
L(s)  = 1  − 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.999·5-s + 0.660·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s + 1.14·10-s − 0.178·12-s − 0.0654·13-s + 0.432·14-s + 0.577·15-s − 1.21·16-s − 0.485·17-s − 0.381·18-s + 0.337·19-s − 0.309·20-s + 0.218·21-s − 0.417·23-s − 0.456·24-s + 0.0749·26-s − 0.192·27-s − 0.116·28-s − 0.928·29-s − 0.660·30-s + 1.88·31-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  =  1
Selberg data  =  $(2,\ 2541,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
13 \( 1 + 0.236T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 1.47T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + 1.47T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 - 8.47T + 43T^{2} \)
47 \( 1 - 9.47T + 47T^{2} \)
53 \( 1 - 6.47T + 53T^{2} \)
59 \( 1 - 7.94T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 - 9.18T + 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 0.472T + 79T^{2} \)
83 \( 1 + 7.52T + 83T^{2} \)
89 \( 1 + 0.472T + 89T^{2} \)
97 \( 1 - 9.41T + 97T^{2} \)
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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

−8.514288415388442860813879361933, −7.81159807287467294909904150150, −7.24693812801127627137230282782, −6.46563243408670096258286957513, −5.43795392521933034685995421633, −4.41126713093883700683468618247, −3.82755163107651186547127156095, −2.41239918951327886656515781669, −1.00021499715164530701316714837, 0, 1.00021499715164530701316714837, 2.41239918951327886656515781669, 3.82755163107651186547127156095, 4.41126713093883700683468618247, 5.43795392521933034685995421633, 6.46563243408670096258286957513, 7.24693812801127627137230282782, 7.81159807287467294909904150150, 8.514288415388442860813879361933

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