Properties

Degree 2
Conductor $ 5 \cdot 11 \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.82·2-s + 0.572·3-s + 1.33·4-s + 5-s − 1.04·6-s − 3.97·7-s + 1.21·8-s − 2.67·9-s − 1.82·10-s + 11-s + 0.762·12-s − 5.95·13-s + 7.26·14-s + 0.572·15-s − 4.88·16-s + 0.0624·17-s + 4.87·18-s + 7.47·19-s + 1.33·20-s − 2.27·21-s − 1.82·22-s − 4.78·23-s + 0.698·24-s + 25-s + 10.8·26-s − 3.24·27-s − 5.29·28-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.330·3-s + 0.666·4-s + 0.447·5-s − 0.426·6-s − 1.50·7-s + 0.431·8-s − 0.890·9-s − 0.577·10-s + 0.301·11-s + 0.220·12-s − 1.65·13-s + 1.94·14-s + 0.147·15-s − 1.22·16-s + 0.0151·17-s + 1.14·18-s + 1.71·19-s + 0.297·20-s − 0.496·21-s − 0.389·22-s − 0.997·23-s + 0.142·24-s + 0.200·25-s + 2.13·26-s − 0.624·27-s − 1.00·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4015\)    =    \(5 \cdot 11 \cdot 73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4015} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4474637518$
$L(\frac12)$  $\approx$  $0.4474637518$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11,\;73\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;11,\;73\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - T \)
11 \( 1 - T \)
73 \( 1 - T \)
good2 \( 1 + 1.82T + 2T^{2} \)
3 \( 1 - 0.572T + 3T^{2} \)
7 \( 1 + 3.97T + 7T^{2} \)
13 \( 1 + 5.95T + 13T^{2} \)
17 \( 1 - 0.0624T + 17T^{2} \)
19 \( 1 - 7.47T + 19T^{2} \)
23 \( 1 + 4.78T + 23T^{2} \)
29 \( 1 - 0.732T + 29T^{2} \)
31 \( 1 - 2.17T + 31T^{2} \)
37 \( 1 + 6.76T + 37T^{2} \)
41 \( 1 + 7.30T + 41T^{2} \)
43 \( 1 + 0.257T + 43T^{2} \)
47 \( 1 - 4.16T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 2.87T + 59T^{2} \)
61 \( 1 + 2.48T + 61T^{2} \)
67 \( 1 - 6.77T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
79 \( 1 - 16.8T + 79T^{2} \)
83 \( 1 - 4.36T + 83T^{2} \)
89 \( 1 + 7.39T + 89T^{2} \)
97 \( 1 + 6.04T + 97T^{2} \)
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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

−8.608921389679276695066371762287, −7.80878152021425302052567030251, −7.17626711709780701352216216326, −6.51014543659030960854084146169, −5.62293471580541313513812521491, −4.79013065826204228480374466325, −3.46328132194273898764325361326, −2.80502733781917098767417993538, −1.87000784452086517936768003791, −0.43733790015470598813474193925, 0.43733790015470598813474193925, 1.87000784452086517936768003791, 2.80502733781917098767417993538, 3.46328132194273898764325361326, 4.79013065826204228480374466325, 5.62293471580541313513812521491, 6.51014543659030960854084146169, 7.17626711709780701352216216326, 7.80878152021425302052567030251, 8.608921389679276695066371762287

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