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  + 2.39·2-s + 3.71·4-s + 5-s + 1.95·7-s + 4.09·8-s + 2.39·10-s + 4.29·11-s + 1.05·13-s + 4.67·14-s + 2.36·16-s − 3.65·17-s − 3.07·19-s + 3.71·20-s + 10.2·22-s + 1.46·23-s + 25-s + 2.51·26-s + 7.25·28-s + 8.44·29-s + 4.24·31-s − 2.54·32-s − 8.73·34-s + 1.95·35-s − 4.61·37-s − 7.35·38-s + 4.09·40-s − 0.785·41-s + ⋯
L(s)  = 1  + 1.69·2-s + 1.85·4-s + 0.447·5-s + 0.738·7-s + 1.44·8-s + 0.755·10-s + 1.29·11-s + 0.292·13-s + 1.24·14-s + 0.590·16-s − 0.886·17-s − 0.706·19-s + 0.830·20-s + 2.18·22-s + 0.305·23-s + 0.200·25-s + 0.493·26-s + 1.37·28-s + 1.56·29-s + 0.761·31-s − 0.449·32-s − 1.49·34-s + 0.330·35-s − 0.758·37-s − 1.19·38-s + 0.647·40-s − 0.122·41-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  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.838509254$
$L(\frac12)$  $\approx$  $6.838509254$
$L(\frac{3}{2})$   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 \( 1 - 2.39T + 2T^{2} \)
7 \( 1 - 1.95T + 7T^{2} \)
11 \( 1 - 4.29T + 11T^{2} \)
13 \( 1 - 1.05T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 + 3.07T + 19T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 - 8.44T + 29T^{2} \)
31 \( 1 - 4.24T + 31T^{2} \)
37 \( 1 + 4.61T + 37T^{2} \)
41 \( 1 + 0.785T + 41T^{2} \)
43 \( 1 + 3.60T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 9.07T + 53T^{2} \)
59 \( 1 - 5.36T + 59T^{2} \)
61 \( 1 + 5.59T + 61T^{2} \)
67 \( 1 + 7.92T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 2.47T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
97 \( 1 - 5.96T + 97T^{2} \)
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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

−8.631743654010530377106866067796, −7.31384729028771310471710090547, −6.61800666850569834725955469567, −6.19118561416418827914855255884, −5.35414523167311294253984538633, −4.48221434061278478404093098781, −4.19968384219659426395014150928, −3.11695519039676024657672729734, −2.24257428578013519286376568567, −1.33689810125117329232002439819, 1.33689810125117329232002439819, 2.24257428578013519286376568567, 3.11695519039676024657672729734, 4.19968384219659426395014150928, 4.48221434061278478404093098781, 5.35414523167311294253984538633, 6.19118561416418827914855255884, 6.61800666850569834725955469567, 7.31384729028771310471710090547, 8.631743654010530377106866067796

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