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  + 2.16·2-s − 1.55·3-s + 2.69·4-s + 5-s − 3.36·6-s + 1.94·7-s + 1.50·8-s − 0.592·9-s + 2.16·10-s + 11-s − 4.18·12-s + 4.81·13-s + 4.22·14-s − 1.55·15-s − 2.12·16-s + 4.21·17-s − 1.28·18-s − 4.40·19-s + 2.69·20-s − 3.02·21-s + 2.16·22-s − 1.88·23-s − 2.33·24-s + 25-s + 10.4·26-s + 5.57·27-s + 5.25·28-s + ⋯
L(s)  = 1  + 1.53·2-s − 0.895·3-s + 1.34·4-s + 0.447·5-s − 1.37·6-s + 0.737·7-s + 0.532·8-s − 0.197·9-s + 0.685·10-s + 0.301·11-s − 1.20·12-s + 1.33·13-s + 1.12·14-s − 0.400·15-s − 0.531·16-s + 1.02·17-s − 0.302·18-s − 1.01·19-s + 0.602·20-s − 0.660·21-s + 0.461·22-s − 0.392·23-s − 0.476·24-s + 0.200·25-s + 2.04·26-s + 1.07·27-s + 0.993·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.108454554$
$L(\frac12)$  $\approx$  $4.108454554$
$L(\frac{3}{2})$   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 \( 1 - 2.16T + 2T^{2} \)
3 \( 1 + 1.55T + 3T^{2} \)
7 \( 1 - 1.94T + 7T^{2} \)
13 \( 1 - 4.81T + 13T^{2} \)
17 \( 1 - 4.21T + 17T^{2} \)
19 \( 1 + 4.40T + 19T^{2} \)
23 \( 1 + 1.88T + 23T^{2} \)
29 \( 1 - 7.25T + 29T^{2} \)
31 \( 1 + 7.65T + 31T^{2} \)
37 \( 1 - 8.64T + 37T^{2} \)
41 \( 1 + 2.23T + 41T^{2} \)
43 \( 1 - 9.48T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 3.93T + 53T^{2} \)
59 \( 1 - 1.70T + 59T^{2} \)
61 \( 1 - 4.40T + 61T^{2} \)
67 \( 1 + 2.87T + 67T^{2} \)
71 \( 1 + 6.41T + 71T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 - 0.759T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−8.464967017662376350243874643472, −7.41220447458418379234881332079, −6.48411280609378224628563100192, −5.80299173588672433497871883727, −5.72710581945276805356611914484, −4.67674129349942895395556782041, −4.12832851355355777165518493337, −3.19433119245943952508325058399, −2.18372797222680687606226940247, −1.01354274993120095396128963810, 1.01354274993120095396128963810, 2.18372797222680687606226940247, 3.19433119245943952508325058399, 4.12832851355355777165518493337, 4.67674129349942895395556782041, 5.72710581945276805356611914484, 5.80299173588672433497871883727, 6.48411280609378224628563100192, 7.41220447458418379234881332079, 8.464967017662376350243874643472

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