Properties

Degree 2
Conductor $ 3 \cdot 13 \cdot 103 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.59·2-s − 3-s + 0.549·4-s − 3.22·5-s − 1.59·6-s + 2.57·7-s − 2.31·8-s + 9-s − 5.14·10-s − 3.57·11-s − 0.549·12-s + 13-s + 4.11·14-s + 3.22·15-s − 4.79·16-s + 6.66·17-s + 1.59·18-s − 5.80·19-s − 1.77·20-s − 2.57·21-s − 5.71·22-s − 1.56·23-s + 2.31·24-s + 5.38·25-s + 1.59·26-s − 27-s + 1.41·28-s + ⋯
L(s)  = 1  + 1.12·2-s − 0.577·3-s + 0.274·4-s − 1.44·5-s − 0.651·6-s + 0.974·7-s − 0.818·8-s + 0.333·9-s − 1.62·10-s − 1.07·11-s − 0.158·12-s + 0.277·13-s + 1.10·14-s + 0.832·15-s − 1.19·16-s + 1.61·17-s + 0.376·18-s − 1.33·19-s − 0.395·20-s − 0.562·21-s − 1.21·22-s − 0.327·23-s + 0.472·24-s + 1.07·25-s + 0.313·26-s − 0.192·27-s + 0.267·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4017\)    =    \(3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4017} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.482621082$
$L(\frac12)$  $\approx$  $1.482621082$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13,\;103\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 + T \)
good2 \( 1 - 1.59T + 2T^{2} \)
5 \( 1 + 3.22T + 5T^{2} \)
7 \( 1 - 2.57T + 7T^{2} \)
11 \( 1 + 3.57T + 11T^{2} \)
17 \( 1 - 6.66T + 17T^{2} \)
19 \( 1 + 5.80T + 19T^{2} \)
23 \( 1 + 1.56T + 23T^{2} \)
29 \( 1 + 3.85T + 29T^{2} \)
31 \( 1 + 2.09T + 31T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 - 8.46T + 41T^{2} \)
43 \( 1 + 4.32T + 43T^{2} \)
47 \( 1 - 8.30T + 47T^{2} \)
53 \( 1 - 5.20T + 53T^{2} \)
59 \( 1 - 7.94T + 59T^{2} \)
61 \( 1 + 0.135T + 61T^{2} \)
67 \( 1 - 6.03T + 67T^{2} \)
71 \( 1 - 6.61T + 71T^{2} \)
73 \( 1 + 6.56T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 3.61T + 83T^{2} \)
89 \( 1 - 1.01T + 89T^{2} \)
97 \( 1 - 7.20T + 97T^{2} \)
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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

−8.143932079904375216016564515703, −7.76318240169967539670459933953, −6.95088277851109076863538725779, −5.85356617257149102707100747921, −5.35422082187943559112301654396, −4.64820252448424795792250921599, −3.98790420633237920677341584326, −3.39954202108686956003718405072, −2.19468988484635196237994015749, −0.59001572515744252675893387014, 0.59001572515744252675893387014, 2.19468988484635196237994015749, 3.39954202108686956003718405072, 3.98790420633237920677341584326, 4.64820252448424795792250921599, 5.35422082187943559112301654396, 5.85356617257149102707100747921, 6.95088277851109076863538725779, 7.76318240169967539670459933953, 8.143932079904375216016564515703

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