Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.22·2-s − 3-s + 2.92·4-s − 0.165·5-s + 2.22·6-s − 2.06·8-s + 9-s + 0.368·10-s + 2.45·11-s − 2.92·12-s − 0.862·13-s + 0.165·15-s − 1.27·16-s − 4.02·17-s − 2.22·18-s − 2.72·19-s − 0.486·20-s − 5.43·22-s − 4.73·23-s + 2.06·24-s − 4.97·25-s + 1.91·26-s − 27-s − 6.41·29-s − 0.368·30-s − 4.79·31-s + 6.96·32-s + ⋯
L(s)  = 1  − 1.56·2-s − 0.577·3-s + 1.46·4-s − 0.0742·5-s + 0.906·6-s − 0.729·8-s + 0.333·9-s + 0.116·10-s + 0.738·11-s − 0.845·12-s − 0.239·13-s + 0.0428·15-s − 0.319·16-s − 0.975·17-s − 0.523·18-s − 0.625·19-s − 0.108·20-s − 1.15·22-s − 0.988·23-s + 0.421·24-s − 0.994·25-s + 0.375·26-s − 0.192·27-s − 1.19·29-s − 0.0672·30-s − 0.862·31-s + 1.23·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3086774315$
$L(\frac12)$  $\approx$  $0.3086774315$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 + 2.22T + 2T^{2} \)
5 \( 1 + 0.165T + 5T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 + 0.862T + 13T^{2} \)
17 \( 1 + 4.02T + 17T^{2} \)
19 \( 1 + 2.72T + 19T^{2} \)
23 \( 1 + 4.73T + 23T^{2} \)
29 \( 1 + 6.41T + 29T^{2} \)
31 \( 1 + 4.79T + 31T^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
43 \( 1 - 7.56T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 6.13T + 53T^{2} \)
59 \( 1 + 8.52T + 59T^{2} \)
61 \( 1 - 0.773T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 4.99T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 5.29T + 83T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−8.145949427664635177263989042134, −7.48131718249941956598630682850, −6.81706666041392803083591203792, −6.26830555229923190210478010597, −5.41572943679356454307843934597, −4.36481439306826860958197789840, −3.69431339548009053371534647672, −2.16827599532079230008708779222, −1.70636355357256697438916742049, −0.37607161946350356870513374068, 0.37607161946350356870513374068, 1.70636355357256697438916742049, 2.16827599532079230008708779222, 3.69431339548009053371534647672, 4.36481439306826860958197789840, 5.41572943679356454307843934597, 6.26830555229923190210478010597, 6.81706666041392803083591203792, 7.48131718249941956598630682850, 8.145949427664635177263989042134

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