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  + 2.18·2-s + 1.24·3-s + 2.78·4-s + 5-s + 2.71·6-s + 2.13·7-s + 1.71·8-s − 1.45·9-s + 2.18·10-s + 11-s + 3.45·12-s + 5.05·13-s + 4.67·14-s + 1.24·15-s − 1.81·16-s − 1.12·17-s − 3.18·18-s − 0.136·19-s + 2.78·20-s + 2.65·21-s + 2.18·22-s + 1.74·23-s + 2.13·24-s + 25-s + 11.0·26-s − 5.53·27-s + 5.95·28-s + ⋯
L(s)  = 1  + 1.54·2-s + 0.717·3-s + 1.39·4-s + 0.447·5-s + 1.10·6-s + 0.807·7-s + 0.607·8-s − 0.485·9-s + 0.691·10-s + 0.301·11-s + 0.998·12-s + 1.40·13-s + 1.24·14-s + 0.320·15-s − 0.452·16-s − 0.273·17-s − 0.751·18-s − 0.0313·19-s + 0.622·20-s + 0.579·21-s + 0.466·22-s + 0.364·23-s + 0.435·24-s + 0.200·25-s + 2.16·26-s − 1.06·27-s + 1.12·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.189253306$
$L(\frac12)$  $\approx$  $7.189253306$
$L(\frac{3}{2})$   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 \( 1 - 2.18T + 2T^{2} \)
3 \( 1 - 1.24T + 3T^{2} \)
7 \( 1 - 2.13T + 7T^{2} \)
13 \( 1 - 5.05T + 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 + 0.136T + 19T^{2} \)
23 \( 1 - 1.74T + 23T^{2} \)
29 \( 1 - 6.11T + 29T^{2} \)
31 \( 1 - 4.25T + 31T^{2} \)
37 \( 1 - 1.77T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 4.30T + 43T^{2} \)
47 \( 1 + 6.68T + 47T^{2} \)
53 \( 1 + 5.98T + 53T^{2} \)
59 \( 1 - 7.89T + 59T^{2} \)
61 \( 1 + 7.11T + 61T^{2} \)
67 \( 1 + 4.15T + 67T^{2} \)
71 \( 1 - 7.05T + 71T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 15.9T + 83T^{2} \)
89 \( 1 - 6.06T + 89T^{2} \)
97 \( 1 + 1.06T + 97T^{2} \)
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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

−8.443076591645658971612170613837, −7.73620562895260276541018954311, −6.52131266957348647140216615819, −6.21074385235960561421281253157, −5.32474699309294406811083168833, −4.63588743958808166756045987646, −3.86607212097415204782515617019, −3.09037008168707088216416484191, −2.38669004000101613006047781036, −1.34722721862104729207894354044, 1.34722721862104729207894354044, 2.38669004000101613006047781036, 3.09037008168707088216416484191, 3.86607212097415204782515617019, 4.63588743958808166756045987646, 5.32474699309294406811083168833, 6.21074385235960561421281253157, 6.52131266957348647140216615819, 7.73620562895260276541018954311, 8.443076591645658971612170613837

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