Properties

Degree 2
Conductor $ 2 \cdot 23 \cdot 131 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.37·3-s + 4-s + 3.53·5-s + 2.37·6-s − 2.86·7-s − 8-s + 2.62·9-s − 3.53·10-s − 4.60·11-s − 2.37·12-s + 2.65·13-s + 2.86·14-s − 8.37·15-s + 16-s − 6.58·17-s − 2.62·18-s + 1.84·19-s + 3.53·20-s + 6.79·21-s + 4.60·22-s + 23-s + 2.37·24-s + 7.46·25-s − 2.65·26-s + 0.885·27-s − 2.86·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.36·3-s + 0.5·4-s + 1.57·5-s + 0.968·6-s − 1.08·7-s − 0.353·8-s + 0.875·9-s − 1.11·10-s − 1.38·11-s − 0.684·12-s + 0.737·13-s + 0.765·14-s − 2.16·15-s + 0.250·16-s − 1.59·17-s − 0.619·18-s + 0.422·19-s + 0.789·20-s + 1.48·21-s + 0.981·22-s + 0.208·23-s + 0.484·24-s + 1.49·25-s − 0.521·26-s + 0.170·27-s − 0.541·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6026\)    =    \(2 \cdot 23 \cdot 131\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6026} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6026,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;23,\;131\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;23,\;131\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
23 \( 1 - T \)
131 \( 1 - T \)
good3 \( 1 + 2.37T + 3T^{2} \)
5 \( 1 - 3.53T + 5T^{2} \)
7 \( 1 + 2.86T + 7T^{2} \)
11 \( 1 + 4.60T + 11T^{2} \)
13 \( 1 - 2.65T + 13T^{2} \)
17 \( 1 + 6.58T + 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
29 \( 1 - 2.39T + 29T^{2} \)
31 \( 1 + 6.52T + 31T^{2} \)
37 \( 1 - 3.71T + 37T^{2} \)
41 \( 1 - 0.349T + 41T^{2} \)
43 \( 1 - 5.24T + 43T^{2} \)
47 \( 1 - 13.5T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 + 2.43T + 59T^{2} \)
61 \( 1 - 8.63T + 61T^{2} \)
67 \( 1 + 7.42T + 67T^{2} \)
71 \( 1 + 1.64T + 71T^{2} \)
73 \( 1 + 4.60T + 73T^{2} \)
79 \( 1 - 7.32T + 79T^{2} \)
83 \( 1 + 7.98T + 83T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + 4.38T + 97T^{2} \)
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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

−7.52145774080277926266341651562, −6.75122579529140199253470530248, −6.33128862294581488588766514757, −5.59173374286635480020711591344, −5.40173307293539489739106919691, −4.19637820530235045071988597550, −2.81120870906001662895955730724, −2.25614432664899981493171716363, −1.02760729263460372720102465612, 0, 1.02760729263460372720102465612, 2.25614432664899981493171716363, 2.81120870906001662895955730724, 4.19637820530235045071988597550, 5.40173307293539489739106919691, 5.59173374286635480020711591344, 6.33128862294581488588766514757, 6.75122579529140199253470530248, 7.52145774080277926266341651562

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