Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 13 \cdot 31 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 5-s − 7-s + 9-s − 3·11-s − 2·12-s + 13-s − 15-s + 4·16-s − 3·17-s + 2·19-s + 2·20-s − 21-s + 3·23-s + 25-s + 27-s + 2·28-s + 31-s − 3·33-s + 35-s − 2·36-s − 7·37-s + 39-s − 9·41-s − 10·43-s + 6·44-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s + 0.277·13-s − 0.258·15-s + 16-s − 0.727·17-s + 0.458·19-s + 0.447·20-s − 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.179·31-s − 0.522·33-s + 0.169·35-s − 1/3·36-s − 1.15·37-s + 0.160·39-s − 1.40·41-s − 1.52·43-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6045} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6045,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.134212625\)
\(L(\frac12)\)  \(\approx\)  \(1.134212625\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;13,\;31\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 - T \)
31 \( 1 - T \)
good2 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−8.329505882528099456631445275243, −7.42903980837140067522789440470, −6.87245909228288937763530243911, −5.81863293005061305975785369247, −5.01415283154477736879111848822, −4.48345379476440159851870151791, −3.48633135418296489890274782867, −3.10094354832761761562446618161, −1.86463862240861561376209124835, −0.53811607694792309300370208095, 0.53811607694792309300370208095, 1.86463862240861561376209124835, 3.10094354832761761562446618161, 3.48633135418296489890274782867, 4.48345379476440159851870151791, 5.01415283154477736879111848822, 5.81863293005061305975785369247, 6.87245909228288937763530243911, 7.42903980837140067522789440470, 8.329505882528099456631445275243

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