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  − 0.162·2-s − 1.97·4-s + 5-s − 0.475·7-s + 0.644·8-s − 0.162·10-s + 5.19·11-s − 1.30·13-s + 0.0770·14-s + 3.84·16-s + 7.40·17-s + 6.83·19-s − 1.97·20-s − 0.842·22-s − 2.69·23-s + 25-s + 0.211·26-s + 0.938·28-s + 0.00689·29-s − 8.54·31-s − 1.91·32-s − 1.20·34-s − 0.475·35-s − 4.56·37-s − 1.10·38-s + 0.644·40-s − 3.41·41-s + ⋯
L(s)  = 1  − 0.114·2-s − 0.986·4-s + 0.447·5-s − 0.179·7-s + 0.227·8-s − 0.0512·10-s + 1.56·11-s − 0.361·13-s + 0.0205·14-s + 0.960·16-s + 1.79·17-s + 1.56·19-s − 0.441·20-s − 0.179·22-s − 0.561·23-s + 0.200·25-s + 0.0414·26-s + 0.177·28-s + 0.00128·29-s − 1.53·31-s − 0.337·32-s − 0.205·34-s − 0.0803·35-s − 0.749·37-s − 0.179·38-s + 0.101·40-s − 0.532·41-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  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.793363889$
$L(\frac12)$  $\approx$  $1.793363889$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 - T \)
89 \( 1 - T \)
good2 \( 1 + 0.162T + 2T^{2} \)
7 \( 1 + 0.475T + 7T^{2} \)
11 \( 1 - 5.19T + 11T^{2} \)
13 \( 1 + 1.30T + 13T^{2} \)
17 \( 1 - 7.40T + 17T^{2} \)
19 \( 1 - 6.83T + 19T^{2} \)
23 \( 1 + 2.69T + 23T^{2} \)
29 \( 1 - 0.00689T + 29T^{2} \)
31 \( 1 + 8.54T + 31T^{2} \)
37 \( 1 + 4.56T + 37T^{2} \)
41 \( 1 + 3.41T + 41T^{2} \)
43 \( 1 - 9.87T + 43T^{2} \)
47 \( 1 - 4.62T + 47T^{2} \)
53 \( 1 + 2.86T + 53T^{2} \)
59 \( 1 - 1.13T + 59T^{2} \)
61 \( 1 - 7.30T + 61T^{2} \)
67 \( 1 + 1.79T + 67T^{2} \)
71 \( 1 + 2.91T + 71T^{2} \)
73 \( 1 + 2.26T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 6.35T + 83T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−8.685258833351646013211737605205, −7.61016742271768827804789243786, −7.21565725695907457788414804255, −5.98628287630979239134620388049, −5.55851865590585429212043660524, −4.72376620810705676125894119643, −3.70355642119206810020935302391, −3.28145456542307878505249676004, −1.68489229668470201805370798918, −0.856155226417646382151279930522, 0.856155226417646382151279930522, 1.68489229668470201805370798918, 3.28145456542307878505249676004, 3.70355642119206810020935302391, 4.72376620810705676125894119643, 5.55851865590585429212043660524, 5.98628287630979239134620388049, 7.21565725695907457788414804255, 7.61016742271768827804789243786, 8.685258833351646013211737605205

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