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  + 2.61·2-s + 0.492·3-s + 4.82·4-s + 5-s + 1.28·6-s + 3.47·7-s + 7.37·8-s − 2.75·9-s + 2.61·10-s + 11-s + 2.37·12-s − 0.698·13-s + 9.06·14-s + 0.492·15-s + 9.61·16-s + 6.56·17-s − 7.20·18-s + 3.10·19-s + 4.82·20-s + 1.70·21-s + 2.61·22-s − 3.07·23-s + 3.62·24-s + 25-s − 1.82·26-s − 2.83·27-s + 16.7·28-s + ⋯
L(s)  = 1  + 1.84·2-s + 0.284·3-s + 2.41·4-s + 0.447·5-s + 0.524·6-s + 1.31·7-s + 2.60·8-s − 0.919·9-s + 0.825·10-s + 0.301·11-s + 0.685·12-s − 0.193·13-s + 2.42·14-s + 0.127·15-s + 2.40·16-s + 1.59·17-s − 1.69·18-s + 0.711·19-s + 1.07·20-s + 0.372·21-s + 0.556·22-s − 0.641·23-s + 0.740·24-s + 0.200·25-s − 0.357·26-s − 0.545·27-s + 3.16·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.553402377$
$L(\frac12)$  $\approx$  $8.553402377$
$L(\frac{3}{2})$   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 \( 1 - 2.61T + 2T^{2} \)
3 \( 1 - 0.492T + 3T^{2} \)
7 \( 1 - 3.47T + 7T^{2} \)
13 \( 1 + 0.698T + 13T^{2} \)
17 \( 1 - 6.56T + 17T^{2} \)
19 \( 1 - 3.10T + 19T^{2} \)
23 \( 1 + 3.07T + 23T^{2} \)
29 \( 1 + 4.74T + 29T^{2} \)
31 \( 1 + 8.90T + 31T^{2} \)
37 \( 1 + 1.42T + 37T^{2} \)
41 \( 1 + 5.91T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 0.200T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + 7.84T + 59T^{2} \)
61 \( 1 + 8.96T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 + 4.05T + 71T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−8.097527791371719315593830721242, −7.61605187740950412096744014272, −6.78800958411912719886075285840, −5.73865344907480685476372432032, −5.43185446663838081540266789916, −4.87939007442420308118889335977, −3.73105646416311990019974138693, −3.28822347927592567070381431097, −2.18622396018091752165090325237, −1.54059595468405124647464467684, 1.54059595468405124647464467684, 2.18622396018091752165090325237, 3.28822347927592567070381431097, 3.73105646416311990019974138693, 4.87939007442420308118889335977, 5.43185446663838081540266789916, 5.73865344907480685476372432032, 6.78800958411912719886075285840, 7.61605187740950412096744014272, 8.097527791371719315593830721242

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