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  − 0.742·2-s + 0.782·3-s − 1.44·4-s + 5-s − 0.580·6-s + 0.262·7-s + 2.56·8-s − 2.38·9-s − 0.742·10-s + 11-s − 1.13·12-s − 1.63·13-s − 0.194·14-s + 0.782·15-s + 0.996·16-s − 3.94·17-s + 1.77·18-s + 3.18·19-s − 1.44·20-s + 0.205·21-s − 0.742·22-s − 4.37·23-s + 2.00·24-s + 25-s + 1.21·26-s − 4.21·27-s − 0.380·28-s + ⋯
L(s)  = 1  − 0.524·2-s + 0.451·3-s − 0.724·4-s + 0.447·5-s − 0.237·6-s + 0.0991·7-s + 0.905·8-s − 0.795·9-s − 0.234·10-s + 0.301·11-s − 0.327·12-s − 0.452·13-s − 0.0520·14-s + 0.202·15-s + 0.249·16-s − 0.957·17-s + 0.417·18-s + 0.730·19-s − 0.323·20-s + 0.0448·21-s − 0.158·22-s − 0.912·23-s + 0.408·24-s + 0.200·25-s + 0.237·26-s − 0.811·27-s − 0.0718·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.236021749$
$L(\frac12)$  $\approx$  $1.236021749$
$L(\frac{3}{2})$   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 \( 1 + 0.742T + 2T^{2} \)
3 \( 1 - 0.782T + 3T^{2} \)
7 \( 1 - 0.262T + 7T^{2} \)
13 \( 1 + 1.63T + 13T^{2} \)
17 \( 1 + 3.94T + 17T^{2} \)
19 \( 1 - 3.18T + 19T^{2} \)
23 \( 1 + 4.37T + 23T^{2} \)
29 \( 1 - 4.80T + 29T^{2} \)
31 \( 1 + 0.406T + 31T^{2} \)
37 \( 1 - 6.86T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 9.75T + 43T^{2} \)
47 \( 1 - 7.88T + 47T^{2} \)
53 \( 1 + 3.45T + 53T^{2} \)
59 \( 1 - 5.96T + 59T^{2} \)
61 \( 1 - 6.28T + 61T^{2} \)
67 \( 1 - 15.0T + 67T^{2} \)
71 \( 1 + 0.367T + 71T^{2} \)
79 \( 1 + 9.69T + 79T^{2} \)
83 \( 1 + 7.65T + 83T^{2} \)
89 \( 1 + 0.317T + 89T^{2} \)
97 \( 1 - 9.10T + 97T^{2} \)
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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

−8.453518481456339581517647916952, −8.020087816963957362786598265717, −7.16676697591224343307590232701, −6.23008547956641758437899813491, −5.45394997757748336808247586599, −4.64830033138282933963750053227, −3.88153268404293562686624319208, −2.81495648781920446086268395711, −1.95299220633293088957856566277, −0.67357227074827834348170915340, 0.67357227074827834348170915340, 1.95299220633293088957856566277, 2.81495648781920446086268395711, 3.88153268404293562686624319208, 4.64830033138282933963750053227, 5.45394997757748336808247586599, 6.23008547956641758437899813491, 7.16676697591224343307590232701, 8.020087816963957362786598265717, 8.453518481456339581517647916952

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