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  + 2.21·2-s + 2.91·4-s − 5-s − 3.75·7-s + 2.02·8-s − 2.21·10-s + 1.54·11-s + 4.15·13-s − 8.31·14-s − 1.33·16-s − 5.74·17-s + 7.03·19-s − 2.91·20-s + 3.41·22-s + 6.77·23-s + 25-s + 9.20·26-s − 10.9·28-s + 10.1·29-s + 0.0578·31-s − 7.01·32-s − 12.7·34-s + 3.75·35-s − 2.18·37-s + 15.5·38-s − 2.02·40-s + 8.82·41-s + ⋯
L(s)  = 1  + 1.56·2-s + 1.45·4-s − 0.447·5-s − 1.41·7-s + 0.716·8-s − 0.700·10-s + 0.464·11-s + 1.15·13-s − 2.22·14-s − 0.334·16-s − 1.39·17-s + 1.61·19-s − 0.651·20-s + 0.728·22-s + 1.41·23-s + 0.200·25-s + 1.80·26-s − 2.06·28-s + 1.88·29-s + 0.0103·31-s − 1.24·32-s − 2.18·34-s + 0.634·35-s − 0.358·37-s + 2.52·38-s − 0.320·40-s + 1.37·41-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  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.923669922$
$L(\frac12)$  $\approx$  $3.923669922$
$L(\frac{3}{2})$   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 \( 1 - 2.21T + 2T^{2} \)
7 \( 1 + 3.75T + 7T^{2} \)
11 \( 1 - 1.54T + 11T^{2} \)
13 \( 1 - 4.15T + 13T^{2} \)
17 \( 1 + 5.74T + 17T^{2} \)
19 \( 1 - 7.03T + 19T^{2} \)
23 \( 1 - 6.77T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 - 0.0578T + 31T^{2} \)
37 \( 1 + 2.18T + 37T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 + 6.08T + 43T^{2} \)
47 \( 1 + 4.08T + 47T^{2} \)
53 \( 1 - 3.49T + 53T^{2} \)
59 \( 1 - 9.20T + 59T^{2} \)
61 \( 1 - 8.32T + 61T^{2} \)
67 \( 1 - 3.93T + 67T^{2} \)
71 \( 1 - 5.49T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 - 8.83T + 79T^{2} \)
83 \( 1 - 9.09T + 83T^{2} \)
97 \( 1 + 2.70T + 97T^{2} \)
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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

−8.548899108007283557911196992467, −7.25491243196850822738821483953, −6.62504763142322047285341611145, −6.32860391915425604926392204972, −5.35340137918502298661634930057, −4.61913186619548827444835039330, −3.74683221530736935911329436194, −3.28075390742662929475083161135, −2.55705563754935317431589997304, −0.904603493956570931623112412073, 0.904603493956570931623112412073, 2.55705563754935317431589997304, 3.28075390742662929475083161135, 3.74683221530736935911329436194, 4.61913186619548827444835039330, 5.35340137918502298661634930057, 6.32860391915425604926392204972, 6.62504763142322047285341611145, 7.25491243196850822738821483953, 8.548899108007283557911196992467

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