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  − 1.45·2-s + 0.119·4-s + 5-s + 3.71·7-s + 2.73·8-s − 1.45·10-s + 1.80·11-s − 6.78·13-s − 5.40·14-s − 4.22·16-s + 1.92·17-s − 6.21·19-s + 0.119·20-s − 2.63·22-s − 1.39·23-s + 25-s + 9.87·26-s + 0.443·28-s + 4.99·29-s + 1.92·31-s + 0.675·32-s − 2.79·34-s + 3.71·35-s + 2.70·37-s + 9.05·38-s + 2.73·40-s − 2.91·41-s + ⋯
L(s)  = 1  − 1.02·2-s + 0.0598·4-s + 0.447·5-s + 1.40·7-s + 0.967·8-s − 0.460·10-s + 0.544·11-s − 1.88·13-s − 1.44·14-s − 1.05·16-s + 0.466·17-s − 1.42·19-s + 0.0267·20-s − 0.560·22-s − 0.291·23-s + 0.200·25-s + 1.93·26-s + 0.0838·28-s + 0.927·29-s + 0.345·31-s + 0.119·32-s − 0.480·34-s + 0.627·35-s + 0.445·37-s + 1.46·38-s + 0.432·40-s − 0.455·41-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  =  $(2,\ 4005,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.175477000$
$L(\frac12)$  $\approx$  $1.175477000$
$L(\frac{3}{2})$   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 \( 1 + 1.45T + 2T^{2} \)
7 \( 1 - 3.71T + 7T^{2} \)
11 \( 1 - 1.80T + 11T^{2} \)
13 \( 1 + 6.78T + 13T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
19 \( 1 + 6.21T + 19T^{2} \)
23 \( 1 + 1.39T + 23T^{2} \)
29 \( 1 - 4.99T + 29T^{2} \)
31 \( 1 - 1.92T + 31T^{2} \)
37 \( 1 - 2.70T + 37T^{2} \)
41 \( 1 + 2.91T + 41T^{2} \)
43 \( 1 - 0.211T + 43T^{2} \)
47 \( 1 - 9.61T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 2.88T + 67T^{2} \)
71 \( 1 - 3.85T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 5.59T + 83T^{2} \)
97 \( 1 - 1.07T + 97T^{2} \)
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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

−8.415202619763697328899038979606, −7.945907854909723539741083623604, −7.23069695411459060404074217696, −6.47909889104213740988821413744, −5.28689867147609591346868374570, −4.74576967549638086628869883419, −4.09011661908282239401677456840, −2.44145063898060468471683883780, −1.85101691958033073978640539281, −0.74525662982879634769340630898, 0.74525662982879634769340630898, 1.85101691958033073978640539281, 2.44145063898060468471683883780, 4.09011661908282239401677456840, 4.74576967549638086628869883419, 5.28689867147609591346868374570, 6.47909889104213740988821413744, 7.23069695411459060404074217696, 7.945907854909723539741083623604, 8.415202619763697328899038979606

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