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  − 2.70·2-s − 1.27·3-s + 5.31·4-s + 5-s + 3.43·6-s − 0.968·7-s − 8.95·8-s − 1.38·9-s − 2.70·10-s + 11-s − 6.75·12-s + 1.17·13-s + 2.61·14-s − 1.27·15-s + 13.5·16-s − 1.19·17-s + 3.73·18-s + 4.63·19-s + 5.31·20-s + 1.23·21-s − 2.70·22-s + 3.13·23-s + 11.3·24-s + 25-s − 3.18·26-s + 5.57·27-s − 5.14·28-s + ⋯
L(s)  = 1  − 1.91·2-s − 0.734·3-s + 2.65·4-s + 0.447·5-s + 1.40·6-s − 0.366·7-s − 3.16·8-s − 0.460·9-s − 0.855·10-s + 0.301·11-s − 1.95·12-s + 0.326·13-s + 0.700·14-s − 0.328·15-s + 3.39·16-s − 0.289·17-s + 0.880·18-s + 1.06·19-s + 1.18·20-s + 0.268·21-s − 0.576·22-s + 0.652·23-s + 2.32·24-s + 0.200·25-s − 0.624·26-s + 1.07·27-s − 0.972·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5295461964$
$L(\frac12)$  $\approx$  $0.5295461964$
$L(\frac{3}{2})$   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 \( 1 + 2.70T + 2T^{2} \)
3 \( 1 + 1.27T + 3T^{2} \)
7 \( 1 + 0.968T + 7T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 + 1.19T + 17T^{2} \)
19 \( 1 - 4.63T + 19T^{2} \)
23 \( 1 - 3.13T + 23T^{2} \)
29 \( 1 - 9.52T + 29T^{2} \)
31 \( 1 + 7.86T + 31T^{2} \)
37 \( 1 + 4.17T + 37T^{2} \)
41 \( 1 - 2.33T + 41T^{2} \)
43 \( 1 + 5.79T + 43T^{2} \)
47 \( 1 - 3.18T + 47T^{2} \)
53 \( 1 + 2.16T + 53T^{2} \)
59 \( 1 - 9.54T + 59T^{2} \)
61 \( 1 - 2.24T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 - 9.15T + 71T^{2} \)
79 \( 1 - 5.23T + 79T^{2} \)
83 \( 1 - 6.06T + 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + 2.72T + 97T^{2} \)
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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

−8.659141571045240602999530554555, −7.87386037511174753065602059722, −6.92632209289088224339117311757, −6.57862792318716793209465568163, −5.80813247561650458596208461687, −5.07061285303226608012345734735, −3.40052831292719600927793527324, −2.62497746782770193079821118549, −1.49690050913096112494870746616, −0.60367772616461096826747229149, 0.60367772616461096826747229149, 1.49690050913096112494870746616, 2.62497746782770193079821118549, 3.40052831292719600927793527324, 5.07061285303226608012345734735, 5.80813247561650458596208461687, 6.57862792318716793209465568163, 6.92632209289088224339117311757, 7.87386037511174753065602059722, 8.659141571045240602999530554555

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