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  − 1.58·2-s − 0.358·3-s + 0.525·4-s + 5-s + 0.569·6-s + 2.77·7-s + 2.34·8-s − 2.87·9-s − 1.58·10-s + 11-s − 0.188·12-s + 1.54·13-s − 4.40·14-s − 0.358·15-s − 4.77·16-s + 3.82·17-s + 4.56·18-s + 5.21·19-s + 0.525·20-s − 0.993·21-s − 1.58·22-s + 4.87·23-s − 0.839·24-s + 25-s − 2.45·26-s + 2.10·27-s + 1.45·28-s + ⋯
L(s)  = 1  − 1.12·2-s − 0.206·3-s + 0.262·4-s + 0.447·5-s + 0.232·6-s + 1.04·7-s + 0.828·8-s − 0.957·9-s − 0.502·10-s + 0.301·11-s − 0.0543·12-s + 0.428·13-s − 1.17·14-s − 0.0925·15-s − 1.19·16-s + 0.926·17-s + 1.07·18-s + 1.19·19-s + 0.117·20-s − 0.216·21-s − 0.338·22-s + 1.01·23-s − 0.171·24-s + 0.200·25-s − 0.481·26-s + 0.404·27-s + 0.275·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.220011384$
$L(\frac12)$  $\approx$  $1.220011384$
$L(\frac{3}{2})$   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 \( 1 + 1.58T + 2T^{2} \)
3 \( 1 + 0.358T + 3T^{2} \)
7 \( 1 - 2.77T + 7T^{2} \)
13 \( 1 - 1.54T + 13T^{2} \)
17 \( 1 - 3.82T + 17T^{2} \)
19 \( 1 - 5.21T + 19T^{2} \)
23 \( 1 - 4.87T + 23T^{2} \)
29 \( 1 + 6.06T + 29T^{2} \)
31 \( 1 - 2.13T + 31T^{2} \)
37 \( 1 - 2.69T + 37T^{2} \)
41 \( 1 + 2.64T + 41T^{2} \)
43 \( 1 - 2.93T + 43T^{2} \)
47 \( 1 - 13.5T + 47T^{2} \)
53 \( 1 + 0.687T + 53T^{2} \)
59 \( 1 + 2.29T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 6.87T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 3.83T + 83T^{2} \)
89 \( 1 + 1.55T + 89T^{2} \)
97 \( 1 - 6.46T + 97T^{2} \)
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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

−8.521508125530951859580375908656, −7.80384117986060243751433509648, −7.36876703670767337243237834133, −6.27270204250110363300827303903, −5.41033577610081734659599367841, −4.95517680173709015523215916258, −3.80110947809252491703810947091, −2.70423466491848019221969468897, −1.54375358264754763489106463518, −0.843532633409597324824027982773, 0.843532633409597324824027982773, 1.54375358264754763489106463518, 2.70423466491848019221969468897, 3.80110947809252491703810947091, 4.95517680173709015523215916258, 5.41033577610081734659599367841, 6.27270204250110363300827303903, 7.36876703670767337243237834133, 7.80384117986060243751433509648, 8.521508125530951859580375908656

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