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  + 1.91·2-s − 0.260·3-s + 1.68·4-s + 5-s − 0.500·6-s − 5.05·7-s − 0.605·8-s − 2.93·9-s + 1.91·10-s + 11-s − 0.439·12-s − 1.62·13-s − 9.71·14-s − 0.260·15-s − 4.53·16-s + 8.05·17-s − 5.62·18-s + 2.76·19-s + 1.68·20-s + 1.31·21-s + 1.91·22-s + 6.01·23-s + 0.157·24-s + 25-s − 3.11·26-s + 1.54·27-s − 8.52·28-s + ⋯
L(s)  = 1  + 1.35·2-s − 0.150·3-s + 0.842·4-s + 0.447·5-s − 0.204·6-s − 1.91·7-s − 0.214·8-s − 0.977·9-s + 0.607·10-s + 0.301·11-s − 0.126·12-s − 0.449·13-s − 2.59·14-s − 0.0673·15-s − 1.13·16-s + 1.95·17-s − 1.32·18-s + 0.633·19-s + 0.376·20-s + 0.287·21-s + 0.409·22-s + 1.25·23-s + 0.0322·24-s + 0.200·25-s − 0.610·26-s + 0.297·27-s − 1.61·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.795715287$
$L(\frac12)$  $\approx$  $2.795715287$
$L(\frac{3}{2})$   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 \( 1 - 1.91T + 2T^{2} \)
3 \( 1 + 0.260T + 3T^{2} \)
7 \( 1 + 5.05T + 7T^{2} \)
13 \( 1 + 1.62T + 13T^{2} \)
17 \( 1 - 8.05T + 17T^{2} \)
19 \( 1 - 2.76T + 19T^{2} \)
23 \( 1 - 6.01T + 23T^{2} \)
29 \( 1 - 9.05T + 29T^{2} \)
31 \( 1 - 3.49T + 31T^{2} \)
37 \( 1 + 2.75T + 37T^{2} \)
41 \( 1 + 2.96T + 41T^{2} \)
43 \( 1 + 8.50T + 43T^{2} \)
47 \( 1 + 1.94T + 47T^{2} \)
53 \( 1 - 8.90T + 53T^{2} \)
59 \( 1 + 8.85T + 59T^{2} \)
61 \( 1 - 8.65T + 61T^{2} \)
67 \( 1 + 4.09T + 67T^{2} \)
71 \( 1 - 0.239T + 71T^{2} \)
79 \( 1 - 0.231T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 + 2.05T + 97T^{2} \)
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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

−8.573693337387852702212017579698, −7.35761660465608100264483582728, −6.51379232983415173221395135672, −6.21421354259469238719968643045, −5.36393378390036052420377073568, −4.94292297879465064072293395022, −3.54269257891913691926005671754, −3.19775260660488191098150951574, −2.62586006901756577301931763033, −0.76437396294742986009579177794, 0.76437396294742986009579177794, 2.62586006901756577301931763033, 3.19775260660488191098150951574, 3.54269257891913691926005671754, 4.94292297879465064072293395022, 5.36393378390036052420377073568, 6.21421354259469238719968643045, 6.51379232983415173221395135672, 7.35761660465608100264483582728, 8.573693337387852702212017579698

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