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  − 0.745·2-s + 3.05·3-s − 1.44·4-s + 5-s − 2.28·6-s − 3.66·7-s + 2.56·8-s + 6.35·9-s − 0.745·10-s + 11-s − 4.41·12-s + 1.02·13-s + 2.72·14-s + 3.05·15-s + 0.974·16-s + 7.92·17-s − 4.73·18-s − 1.53·19-s − 1.44·20-s − 11.1·21-s − 0.745·22-s − 9.02·23-s + 7.85·24-s + 25-s − 0.764·26-s + 10.2·27-s + 5.28·28-s + ⋯
L(s)  = 1  − 0.527·2-s + 1.76·3-s − 0.722·4-s + 0.447·5-s − 0.930·6-s − 1.38·7-s + 0.907·8-s + 2.11·9-s − 0.235·10-s + 0.301·11-s − 1.27·12-s + 0.284·13-s + 0.729·14-s + 0.789·15-s + 0.243·16-s + 1.92·17-s − 1.11·18-s − 0.352·19-s − 0.322·20-s − 2.44·21-s − 0.158·22-s − 1.88·23-s + 1.60·24-s + 0.200·25-s − 0.150·26-s + 1.97·27-s + 0.999·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.384066780$
$L(\frac12)$  $\approx$  $2.384066780$
$L(\frac{3}{2})$   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 \( 1 + 0.745T + 2T^{2} \)
3 \( 1 - 3.05T + 3T^{2} \)
7 \( 1 + 3.66T + 7T^{2} \)
13 \( 1 - 1.02T + 13T^{2} \)
17 \( 1 - 7.92T + 17T^{2} \)
19 \( 1 + 1.53T + 19T^{2} \)
23 \( 1 + 9.02T + 23T^{2} \)
29 \( 1 - 3.19T + 29T^{2} \)
31 \( 1 + 7.54T + 31T^{2} \)
37 \( 1 - 6.07T + 37T^{2} \)
41 \( 1 - 7.65T + 41T^{2} \)
43 \( 1 - 8.29T + 43T^{2} \)
47 \( 1 - 9.63T + 47T^{2} \)
53 \( 1 + 3.09T + 53T^{2} \)
59 \( 1 - 8.32T + 59T^{2} \)
61 \( 1 + 7.95T + 61T^{2} \)
67 \( 1 + 2.85T + 67T^{2} \)
71 \( 1 + 3.17T + 71T^{2} \)
79 \( 1 - 16.2T + 79T^{2} \)
83 \( 1 + 1.77T + 83T^{2} \)
89 \( 1 - 0.725T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−8.612585578718715894993640592402, −7.72455959780799364051325109704, −7.53651111392356940892564310789, −6.29424943013772607204356225553, −5.61691784468099080426598967384, −4.18118509846470484430668375705, −3.76553584267362811256223280931, −2.98954481578456349130131559983, −2.04315561318937665694717649289, −0.914762605455101139972955805406, 0.914762605455101139972955805406, 2.04315561318937665694717649289, 2.98954481578456349130131559983, 3.76553584267362811256223280931, 4.18118509846470484430668375705, 5.61691784468099080426598967384, 6.29424943013772607204356225553, 7.53651111392356940892564310789, 7.72455959780799364051325109704, 8.612585578718715894993640592402

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