Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 89 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.584·2-s − 1.65·4-s − 5-s − 2.74·7-s + 2.13·8-s + 0.584·10-s + 0.429·11-s − 6.91·13-s + 1.60·14-s + 2.06·16-s − 5.29·17-s − 1.63·19-s + 1.65·20-s − 0.251·22-s + 2.26·23-s + 25-s + 4.04·26-s + 4.55·28-s − 3.93·29-s − 2.28·31-s − 5.48·32-s + 3.09·34-s + 2.74·35-s − 7.74·37-s + 0.958·38-s − 2.13·40-s + 5.18·41-s + ⋯
L(s)  = 1  − 0.413·2-s − 0.828·4-s − 0.447·5-s − 1.03·7-s + 0.756·8-s + 0.184·10-s + 0.129·11-s − 1.91·13-s + 0.429·14-s + 0.515·16-s − 1.28·17-s − 0.375·19-s + 0.370·20-s − 0.0535·22-s + 0.473·23-s + 0.200·25-s + 0.793·26-s + 0.860·28-s − 0.731·29-s − 0.409·31-s − 0.969·32-s + 0.530·34-s + 0.464·35-s − 1.27·37-s + 0.155·38-s − 0.338·40-s + 0.810·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4005} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1781088063$
$L(\frac12)$  $\approx$  $0.1781088063$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;89\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;89\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
89 \( 1 + T \)
good2 \( 1 + 0.584T + 2T^{2} \)
7 \( 1 + 2.74T + 7T^{2} \)
11 \( 1 - 0.429T + 11T^{2} \)
13 \( 1 + 6.91T + 13T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 + 1.63T + 19T^{2} \)
23 \( 1 - 2.26T + 23T^{2} \)
29 \( 1 + 3.93T + 29T^{2} \)
31 \( 1 + 2.28T + 31T^{2} \)
37 \( 1 + 7.74T + 37T^{2} \)
41 \( 1 - 5.18T + 41T^{2} \)
43 \( 1 + 6.95T + 43T^{2} \)
47 \( 1 + 8.34T + 47T^{2} \)
53 \( 1 + 5.51T + 53T^{2} \)
59 \( 1 - 6.35T + 59T^{2} \)
61 \( 1 - 9.00T + 61T^{2} \)
67 \( 1 + 8.39T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 5.08T + 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 - 6.64T + 83T^{2} \)
97 \( 1 - 0.828T + 97T^{2} \)
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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

−8.643022257001375121162865685378, −7.69506557447565499727175497577, −7.10182104687381763589053170997, −6.45518979026521459610260087423, −5.25769607758435589754737060380, −4.70092254290167361204300331236, −3.88933756182799797173575566302, −3.01389020421385741416430281131, −1.92367402177552971373473021125, −0.24563271759140740798653463895, 0.24563271759140740798653463895, 1.92367402177552971373473021125, 3.01389020421385741416430281131, 3.88933756182799797173575566302, 4.70092254290167361204300331236, 5.25769607758435589754737060380, 6.45518979026521459610260087423, 7.10182104687381763589053170997, 7.69506557447565499727175497577, 8.643022257001375121162865685378

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