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  − 1.16·2-s − 1.28·3-s − 0.648·4-s + 5-s + 1.48·6-s − 0.641·7-s + 3.07·8-s − 1.35·9-s − 1.16·10-s + 11-s + 0.830·12-s − 2.10·13-s + 0.746·14-s − 1.28·15-s − 2.28·16-s + 0.343·17-s + 1.58·18-s − 5.83·19-s − 0.648·20-s + 0.822·21-s − 1.16·22-s − 6.18·23-s − 3.94·24-s + 25-s + 2.45·26-s + 5.58·27-s + 0.416·28-s + ⋯
L(s)  = 1  − 0.822·2-s − 0.739·3-s − 0.324·4-s + 0.447·5-s + 0.607·6-s − 0.242·7-s + 1.08·8-s − 0.453·9-s − 0.367·10-s + 0.301·11-s + 0.239·12-s − 0.584·13-s + 0.199·14-s − 0.330·15-s − 0.570·16-s + 0.0833·17-s + 0.372·18-s − 1.33·19-s − 0.145·20-s + 0.179·21-s − 0.247·22-s − 1.28·23-s − 0.804·24-s + 0.200·25-s + 0.480·26-s + 1.07·27-s + 0.0786·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4505039786$
$L(\frac12)$  $\approx$  $0.4505039786$
$L(\frac{3}{2})$   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 \( 1 + 1.16T + 2T^{2} \)
3 \( 1 + 1.28T + 3T^{2} \)
7 \( 1 + 0.641T + 7T^{2} \)
13 \( 1 + 2.10T + 13T^{2} \)
17 \( 1 - 0.343T + 17T^{2} \)
19 \( 1 + 5.83T + 19T^{2} \)
23 \( 1 + 6.18T + 23T^{2} \)
29 \( 1 - 7.74T + 29T^{2} \)
31 \( 1 - 2.91T + 31T^{2} \)
37 \( 1 - 0.493T + 37T^{2} \)
41 \( 1 + 8.13T + 41T^{2} \)
43 \( 1 - 3.22T + 43T^{2} \)
47 \( 1 + 8.99T + 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 7.45T + 61T^{2} \)
67 \( 1 + 2.50T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
79 \( 1 - 0.783T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + 2.69T + 89T^{2} \)
97 \( 1 - 9.65T + 97T^{2} \)
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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

−8.362057910701691654171140613558, −8.086160816945672338053589075726, −6.78842696302494737292210678651, −6.42741843991255259457457628143, −5.49412344799903452039071202155, −4.77743134240370962026984202860, −4.03318949671884488923643969010, −2.73635117909004920860910432081, −1.69303147812491358566666725224, −0.45366689420985676690791353788, 0.45366689420985676690791353788, 1.69303147812491358566666725224, 2.73635117909004920860910432081, 4.03318949671884488923643969010, 4.77743134240370962026984202860, 5.49412344799903452039071202155, 6.42741843991255259457457628143, 6.78842696302494737292210678651, 8.086160816945672338053589075726, 8.362057910701691654171140613558

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