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  − 1.99·2-s + 2.62·3-s + 1.99·4-s + 5-s − 5.23·6-s + 2.62·7-s + 0.0113·8-s + 3.86·9-s − 1.99·10-s + 11-s + 5.22·12-s − 1.22·13-s − 5.24·14-s + 2.62·15-s − 4.01·16-s − 6.75·17-s − 7.72·18-s + 5.20·19-s + 1.99·20-s + 6.87·21-s − 1.99·22-s − 0.411·23-s + 0.0296·24-s + 25-s + 2.45·26-s + 2.27·27-s + 5.23·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.51·3-s + 0.997·4-s + 0.447·5-s − 2.13·6-s + 0.991·7-s + 0.00399·8-s + 1.28·9-s − 0.632·10-s + 0.301·11-s + 1.50·12-s − 0.340·13-s − 1.40·14-s + 0.676·15-s − 1.00·16-s − 1.63·17-s − 1.82·18-s + 1.19·19-s + 0.445·20-s + 1.50·21-s − 0.426·22-s − 0.0857·23-s + 0.00604·24-s + 0.200·25-s + 0.480·26-s + 0.437·27-s + 0.988·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.016027020$
$L(\frac12)$  $\approx$  $2.016027020$
$L(\frac{3}{2})$   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 \( 1 + 1.99T + 2T^{2} \)
3 \( 1 - 2.62T + 3T^{2} \)
7 \( 1 - 2.62T + 7T^{2} \)
13 \( 1 + 1.22T + 13T^{2} \)
17 \( 1 + 6.75T + 17T^{2} \)
19 \( 1 - 5.20T + 19T^{2} \)
23 \( 1 + 0.411T + 23T^{2} \)
29 \( 1 - 2.08T + 29T^{2} \)
31 \( 1 + 5.35T + 31T^{2} \)
37 \( 1 + 8.62T + 37T^{2} \)
41 \( 1 - 4.78T + 41T^{2} \)
43 \( 1 - 7.01T + 43T^{2} \)
47 \( 1 - 9.82T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 - 5.13T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + 0.917T + 67T^{2} \)
71 \( 1 - 2.96T + 71T^{2} \)
79 \( 1 - 7.46T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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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

−8.711457662393581622397611034600, −7.924529355059207491651054620794, −7.33683709997526673778752123203, −6.81604116882893966820791378056, −5.46126473530410941109732363062, −4.52919350228542382000482595980, −3.67656064365877400292946580269, −2.32215274802455845920280877370, −2.09852500237912757911947932054, −0.973996650221403173555680637689, 0.973996650221403173555680637689, 2.09852500237912757911947932054, 2.32215274802455845920280877370, 3.67656064365877400292946580269, 4.52919350228542382000482595980, 5.46126473530410941109732363062, 6.81604116882893966820791378056, 7.33683709997526673778752123203, 7.924529355059207491651054620794, 8.711457662393581622397611034600

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