Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.151·2-s − 1.97·4-s + 5-s + 2.88·7-s + 0.604·8-s − 0.151·10-s − 2.75·11-s + 2.78·13-s − 0.438·14-s + 3.86·16-s − 6.24·17-s − 0.721·19-s − 1.97·20-s + 0.418·22-s + 0.0986·23-s + 25-s − 0.423·26-s − 5.70·28-s − 6.61·29-s − 0.885·31-s − 1.79·32-s + 0.949·34-s + 2.88·35-s − 4.03·37-s + 0.109·38-s + 0.604·40-s − 9.13·41-s + ⋯
L(s)  = 1  − 0.107·2-s − 0.988·4-s + 0.447·5-s + 1.09·7-s + 0.213·8-s − 0.0480·10-s − 0.830·11-s + 0.773·13-s − 0.117·14-s + 0.965·16-s − 1.51·17-s − 0.165·19-s − 0.442·20-s + 0.0892·22-s + 0.0205·23-s + 0.200·25-s − 0.0830·26-s − 1.07·28-s − 1.22·29-s − 0.159·31-s − 0.317·32-s + 0.162·34-s + 0.488·35-s − 0.662·37-s + 0.0177·38-s + 0.0955·40-s − 1.42·41-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  =  1
Selberg data  =  $(2,\ 4005,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;89\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 + 0.151T + 2T^{2} \)
7 \( 1 - 2.88T + 7T^{2} \)
11 \( 1 + 2.75T + 11T^{2} \)
13 \( 1 - 2.78T + 13T^{2} \)
17 \( 1 + 6.24T + 17T^{2} \)
19 \( 1 + 0.721T + 19T^{2} \)
23 \( 1 - 0.0986T + 23T^{2} \)
29 \( 1 + 6.61T + 29T^{2} \)
31 \( 1 + 0.885T + 31T^{2} \)
37 \( 1 + 4.03T + 37T^{2} \)
41 \( 1 + 9.13T + 41T^{2} \)
43 \( 1 - 4.57T + 43T^{2} \)
47 \( 1 - 6.64T + 47T^{2} \)
53 \( 1 + 0.765T + 53T^{2} \)
59 \( 1 + 2.79T + 59T^{2} \)
61 \( 1 + 8.50T + 61T^{2} \)
67 \( 1 - 4.23T + 67T^{2} \)
71 \( 1 - 2.60T + 71T^{2} \)
73 \( 1 + 9.47T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−8.178015905718573296743677689293, −7.58496760322654945756537101388, −6.58239763411509047109482376994, −5.64382593502638586461697998359, −5.06479192218535397138150723523, −4.40128456919497806415368592408, −3.57228933279433656146416262399, −2.29161829484172047694955294984, −1.43258806366851688126908271421, 0, 1.43258806366851688126908271421, 2.29161829484172047694955294984, 3.57228933279433656146416262399, 4.40128456919497806415368592408, 5.06479192218535397138150723523, 5.64382593502638586461697998359, 6.58239763411509047109482376994, 7.58496760322654945756537101388, 8.178015905718573296743677689293

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