Properties

Degree 2
Conductor $ 3 \cdot 11^{2} \cdot 17 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 3·5-s + 4·7-s + 9-s − 2·12-s + 13-s + 3·15-s + 4·16-s + 17-s + 19-s − 6·20-s + 4·21-s + 9·23-s + 4·25-s + 27-s − 8·28-s − 6·29-s + 2·31-s + 12·35-s − 2·36-s − 4·37-s + 39-s + 3·41-s + 7·43-s + 3·45-s − 6·47-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1.34·5-s + 1.51·7-s + 1/3·9-s − 0.577·12-s + 0.277·13-s + 0.774·15-s + 16-s + 0.242·17-s + 0.229·19-s − 1.34·20-s + 0.872·21-s + 1.87·23-s + 4/5·25-s + 0.192·27-s − 1.51·28-s − 1.11·29-s + 0.359·31-s + 2.02·35-s − 1/3·36-s − 0.657·37-s + 0.160·39-s + 0.468·41-s + 1.06·43-s + 0.447·45-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6171\)    =    \(3 \cdot 11^{2} \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6171} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6171,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.532526180$
$L(\frac12)$  $\approx$  $3.532526180$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 \)
17 \( 1 - T \)
good2 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−8.123063877468170217097057069948, −7.60195787768679259689608146982, −6.65141393178900011874126957740, −5.62535288746366565403300368914, −5.18754379686639968375671320234, −4.59411129964790589621534187851, −3.66589986463958216194059753670, −2.69595897502735310420716823107, −1.70457407581975008871008195419, −1.09494185617481664600047301525, 1.09494185617481664600047301525, 1.70457407581975008871008195419, 2.69595897502735310420716823107, 3.66589986463958216194059753670, 4.59411129964790589621534187851, 5.18754379686639968375671320234, 5.62535288746366565403300368914, 6.65141393178900011874126957740, 7.60195787768679259689608146982, 8.123063877468170217097057069948

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