Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 79 $
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-s + 3.73·4-s + 3.84·5-s + 2.39·6-s − 2.55·7-s + 4.15·8-s + 9-s + 9.21·10-s + 0.235·11-s + 3.73·12-s + 2.44·13-s − 6.11·14-s + 3.84·15-s + 2.48·16-s + 17-s + 2.39·18-s + 1.83·19-s + 14.3·20-s − 2.55·21-s + 0.563·22-s + 6.80·23-s + 4.15·24-s + 9.80·25-s + 5.86·26-s + 27-s − 9.54·28-s + ⋯
L(s)  = 1  + 1.69·2-s + 0.577·3-s + 1.86·4-s + 1.72·5-s + 0.977·6-s − 0.965·7-s + 1.47·8-s + 0.333·9-s + 2.91·10-s + 0.0709·11-s + 1.07·12-s + 0.679·13-s − 1.63·14-s + 0.993·15-s + 0.621·16-s + 0.242·17-s + 0.564·18-s + 0.420·19-s + 3.21·20-s − 0.557·21-s + 0.120·22-s + 1.41·23-s + 0.848·24-s + 1.96·25-s + 1.15·26-s + 0.192·27-s − 1.80·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4029\)    =    \(3 \cdot 17 \cdot 79\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4029} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4029,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.488553341$
$L(\frac12)$  $\approx$  $8.488553341$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;17,\;79\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;79\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
17 \( 1 - T \)
79 \( 1 - T \)
good2 \( 1 - 2.39T + 2T^{2} \)
5 \( 1 - 3.84T + 5T^{2} \)
7 \( 1 + 2.55T + 7T^{2} \)
11 \( 1 - 0.235T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
19 \( 1 - 1.83T + 19T^{2} \)
23 \( 1 - 6.80T + 23T^{2} \)
29 \( 1 + 6.52T + 29T^{2} \)
31 \( 1 + 2.65T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 + 0.0516T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 - 1.31T + 47T^{2} \)
53 \( 1 + 3.78T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 0.795T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 4.27T + 73T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 - 1.77T + 97T^{2} \)
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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

−8.548023247282513882655338650809, −7.23066889115605674607394539633, −6.65347511127835617244196468287, −6.11703557835532074915929715507, −5.37753538708665564978485629893, −4.88854491265108428409825929019, −3.48844913026272748128205915676, −3.30584585817075707379899359073, −2.29025669655392391859951868552, −1.51592944113259046011961976042, 1.51592944113259046011961976042, 2.29025669655392391859951868552, 3.30584585817075707379899359073, 3.48844913026272748128205915676, 4.88854491265108428409825929019, 5.37753538708665564978485629893, 6.11703557835532074915929715507, 6.65347511127835617244196468287, 7.23066889115605674607394539633, 8.548023247282513882655338650809

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