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  − 1.86·2-s − 2.63·3-s + 1.49·4-s + 5-s + 4.92·6-s − 2.31·7-s + 0.944·8-s + 3.92·9-s − 1.86·10-s + 11-s − 3.93·12-s − 2.05·13-s + 4.33·14-s − 2.63·15-s − 4.75·16-s − 5.64·17-s − 7.34·18-s − 4.69·19-s + 1.49·20-s + 6.10·21-s − 1.86·22-s − 7.33·23-s − 2.48·24-s + 25-s + 3.84·26-s − 2.44·27-s − 3.46·28-s + ⋯
L(s)  = 1  − 1.32·2-s − 1.51·3-s + 0.747·4-s + 0.447·5-s + 2.00·6-s − 0.876·7-s + 0.333·8-s + 1.30·9-s − 0.591·10-s + 0.301·11-s − 1.13·12-s − 0.569·13-s + 1.15·14-s − 0.679·15-s − 1.18·16-s − 1.36·17-s − 1.73·18-s − 1.07·19-s + 0.334·20-s + 1.33·21-s − 0.398·22-s − 1.52·23-s − 0.507·24-s + 0.200·25-s + 0.753·26-s − 0.470·27-s − 0.654·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.08029677088$
$L(\frac12)$  $\approx$  $0.08029677088$
$L(\frac{3}{2})$   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 \( 1 + 1.86T + 2T^{2} \)
3 \( 1 + 2.63T + 3T^{2} \)
7 \( 1 + 2.31T + 7T^{2} \)
13 \( 1 + 2.05T + 13T^{2} \)
17 \( 1 + 5.64T + 17T^{2} \)
19 \( 1 + 4.69T + 19T^{2} \)
23 \( 1 + 7.33T + 23T^{2} \)
29 \( 1 + 5.39T + 29T^{2} \)
31 \( 1 - 2.07T + 31T^{2} \)
37 \( 1 - 2.42T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 0.817T + 43T^{2} \)
47 \( 1 + 2.17T + 47T^{2} \)
53 \( 1 + 8.22T + 53T^{2} \)
59 \( 1 + 0.531T + 59T^{2} \)
61 \( 1 + 6.90T + 61T^{2} \)
67 \( 1 + 6.19T + 67T^{2} \)
71 \( 1 + 8.72T + 71T^{2} \)
79 \( 1 - 2.92T + 79T^{2} \)
83 \( 1 + 7.28T + 83T^{2} \)
89 \( 1 + 2.65T + 89T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−8.596678570944753738088436015109, −7.68462239038853836448382355888, −6.89608622508331053226142709909, −6.29022851728662646478578290515, −5.89045799476414159724235808270, −4.67855218643994933376482341875, −4.15970695568918112618680780974, −2.49440625847153873078316536192, −1.57484530443606877549697352810, −0.21435295700147114082587433837, 0.21435295700147114082587433837, 1.57484530443606877549697352810, 2.49440625847153873078316536192, 4.15970695568918112618680780974, 4.67855218643994933376482341875, 5.89045799476414159724235808270, 6.29022851728662646478578290515, 6.89608622508331053226142709909, 7.68462239038853836448382355888, 8.596678570944753738088436015109

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