Properties

Degree 2
Conductor $ 3 \cdot 29 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 6-s − 7-s + 8-s + 9-s − 11-s − 13-s + 14-s − 16-s − 17-s − 18-s − 21-s + 22-s + 24-s + 25-s + 26-s + 27-s + 29-s − 33-s + 34-s − 39-s + 2·41-s + 42-s − 47-s − 48-s − 50-s + ⋯
L(s)  = 1  − 2-s + 3-s − 6-s − 7-s + 8-s + 9-s − 11-s − 13-s + 14-s − 16-s − 17-s − 18-s − 21-s + 22-s + 24-s + 25-s + 26-s + 27-s + 29-s − 33-s + 34-s − 39-s + 2·41-s + 42-s − 47-s − 48-s − 50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(87\)    =    \(3 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{87} (86, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 87,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.3960826179$
$L(\frac12)$  $\approx$  $0.3960826179$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−14.45102409066779657045769585783, −13.32162862510749184354432270374, −12.66997768785203913828312408994, −10.65716487846662969380774941347, −9.805231985523799033951222248495, −9.004858706215995835885407289726, −7.939979829085170543073834513148, −6.89861709118401082750268414601, −4.60444267458741594170483479006, −2.68368590772453422884570720998, 2.68368590772453422884570720998, 4.60444267458741594170483479006, 6.89861709118401082750268414601, 7.939979829085170543073834513148, 9.004858706215995835885407289726, 9.805231985523799033951222248495, 10.65716487846662969380774941347, 12.66997768785203913828312408994, 13.32162862510749184354432270374, 14.45102409066779657045769585783

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