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  + 2.75·2-s + 5.56·4-s − 5-s + 0.587·7-s + 9.81·8-s − 2.75·10-s + 0.892·11-s − 1.31·13-s + 1.61·14-s + 15.8·16-s + 6.89·17-s + 2.78·19-s − 5.56·20-s + 2.45·22-s + 0.636·23-s + 25-s − 3.61·26-s + 3.27·28-s − 5.55·29-s − 9.03·31-s + 24.0·32-s + 18.9·34-s − 0.587·35-s + 0.632·37-s + 7.66·38-s − 9.81·40-s + 7.36·41-s + ⋯
L(s)  = 1  + 1.94·2-s + 2.78·4-s − 0.447·5-s + 0.222·7-s + 3.47·8-s − 0.869·10-s + 0.268·11-s − 0.364·13-s + 0.432·14-s + 3.96·16-s + 1.67·17-s + 0.639·19-s − 1.24·20-s + 0.523·22-s + 0.132·23-s + 0.200·25-s − 0.709·26-s + 0.618·28-s − 1.03·29-s − 1.62·31-s + 4.24·32-s + 3.25·34-s − 0.0993·35-s + 0.104·37-s + 1.24·38-s − 1.55·40-s + 1.15·41-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  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.350818853$
$L(\frac12)$  $\approx$  $7.350818853$
$L(\frac{3}{2})$   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 \( 1 - 2.75T + 2T^{2} \)
7 \( 1 - 0.587T + 7T^{2} \)
11 \( 1 - 0.892T + 11T^{2} \)
13 \( 1 + 1.31T + 13T^{2} \)
17 \( 1 - 6.89T + 17T^{2} \)
19 \( 1 - 2.78T + 19T^{2} \)
23 \( 1 - 0.636T + 23T^{2} \)
29 \( 1 + 5.55T + 29T^{2} \)
31 \( 1 + 9.03T + 31T^{2} \)
37 \( 1 - 0.632T + 37T^{2} \)
41 \( 1 - 7.36T + 41T^{2} \)
43 \( 1 + 7.41T + 43T^{2} \)
47 \( 1 - 6.16T + 47T^{2} \)
53 \( 1 + 3.18T + 53T^{2} \)
59 \( 1 + 5.68T + 59T^{2} \)
61 \( 1 + 4.98T + 61T^{2} \)
67 \( 1 - 7.78T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
97 \( 1 - 5.86T + 97T^{2} \)
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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

−7.82438969107637365903934714917, −7.62498284069755197868116616093, −6.82015993242991985624632348035, −5.94208521902273935002359649050, −5.31553237274059426741348262246, −4.77996882155707588667059890495, −3.64642158549463268920430725952, −3.48102645695819032440271692892, −2.33994961611655180784545503647, −1.31586027371826108649607455120, 1.31586027371826108649607455120, 2.33994961611655180784545503647, 3.48102645695819032440271692892, 3.64642158549463268920430725952, 4.77996882155707588667059890495, 5.31553237274059426741348262246, 5.94208521902273935002359649050, 6.82015993242991985624632348035, 7.62498284069755197868116616093, 7.82438969107637365903934714917

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