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  − 0.217·2-s − 1.95·4-s + 5-s − 2.89·7-s + 0.857·8-s − 0.217·10-s − 3.07·11-s + 0.0854·13-s + 0.629·14-s + 3.71·16-s + 2.13·17-s + 2.44·19-s − 1.95·20-s + 0.667·22-s − 2.98·23-s + 25-s − 0.0185·26-s + 5.66·28-s + 6.39·29-s + 7.97·31-s − 2.52·32-s − 0.462·34-s − 2.89·35-s − 1.01·37-s − 0.530·38-s + 0.857·40-s + 4.08·41-s + ⋯
L(s)  = 1  − 0.153·2-s − 0.976·4-s + 0.447·5-s − 1.09·7-s + 0.303·8-s − 0.0686·10-s − 0.927·11-s + 0.0237·13-s + 0.168·14-s + 0.929·16-s + 0.517·17-s + 0.560·19-s − 0.436·20-s + 0.142·22-s − 0.622·23-s + 0.200·25-s − 0.00363·26-s + 1.07·28-s + 1.18·29-s + 1.43·31-s − 0.445·32-s − 0.0793·34-s − 0.490·35-s − 0.166·37-s − 0.0860·38-s + 0.135·40-s + 0.637·41-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 \( 1 + 0.217T + 2T^{2} \)
7 \( 1 + 2.89T + 7T^{2} \)
11 \( 1 + 3.07T + 11T^{2} \)
13 \( 1 - 0.0854T + 13T^{2} \)
17 \( 1 - 2.13T + 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
23 \( 1 + 2.98T + 23T^{2} \)
29 \( 1 - 6.39T + 29T^{2} \)
31 \( 1 - 7.97T + 31T^{2} \)
37 \( 1 + 1.01T + 37T^{2} \)
41 \( 1 - 4.08T + 41T^{2} \)
43 \( 1 + 1.92T + 43T^{2} \)
47 \( 1 + 8.73T + 47T^{2} \)
53 \( 1 - 6.19T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 + 4.48T + 67T^{2} \)
71 \( 1 + 9.54T + 71T^{2} \)
73 \( 1 - 1.45T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 4.61T + 83T^{2} \)
97 \( 1 - 2.35T + 97T^{2} \)
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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

−8.193412786732651956361057567627, −7.48466232089848508322647022177, −6.49983720621242584361178127955, −5.84134919383460123193112855848, −5.09617208400192525413286458517, −4.34278728870723357922085654504, −3.30525674727906986117502877489, −2.68011431311991739023121081637, −1.18246191249945642528772407619, 0, 1.18246191249945642528772407619, 2.68011431311991739023121081637, 3.30525674727906986117502877489, 4.34278728870723357922085654504, 5.09617208400192525413286458517, 5.84134919383460123193112855848, 6.49983720621242584361178127955, 7.48466232089848508322647022177, 8.193412786732651956361057567627

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