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 + 3.27·3-s + 4-s + 0.672·5-s − 3.27·6-s − 2.46·7-s − 8-s + 7.74·9-s − 0.672·10-s − 2.72·11-s + 3.27·12-s − 4.85·13-s + 2.46·14-s + 2.20·15-s + 16-s − 5.69·17-s − 7.74·18-s + 4.50·19-s + 0.672·20-s − 8.09·21-s + 2.72·22-s − 23-s − 3.27·24-s − 4.54·25-s + 4.85·26-s + 15.5·27-s − 2.46·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.89·3-s + 0.5·4-s + 0.300·5-s − 1.33·6-s − 0.933·7-s − 0.353·8-s + 2.58·9-s − 0.212·10-s − 0.820·11-s + 0.946·12-s − 1.34·13-s + 0.659·14-s + 0.569·15-s + 0.250·16-s − 1.38·17-s − 1.82·18-s + 1.03·19-s + 0.150·20-s − 1.76·21-s + 0.579·22-s − 0.208·23-s − 0.668·24-s − 0.909·25-s + 0.952·26-s + 2.99·27-s − 0.466·28-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 \( 1 - 3.27T + 3T^{2} \)
5 \( 1 - 0.672T + 5T^{2} \)
7 \( 1 + 2.46T + 7T^{2} \)
11 \( 1 + 2.72T + 11T^{2} \)
13 \( 1 + 4.85T + 13T^{2} \)
17 \( 1 + 5.69T + 17T^{2} \)
19 \( 1 - 4.50T + 19T^{2} \)
29 \( 1 + 0.111T + 29T^{2} \)
31 \( 1 - 0.359T + 31T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + 5.58T + 41T^{2} \)
43 \( 1 + 3.52T + 43T^{2} \)
47 \( 1 - 4.33T + 47T^{2} \)
53 \( 1 - 4.79T + 53T^{2} \)
59 \( 1 + 2.17T + 59T^{2} \)
61 \( 1 + 0.654T + 61T^{2} \)
67 \( 1 + 0.983T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 8.31T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 3.35T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 9.65T + 97T^{2} \)
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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

−7.947439369055605359251871590754, −7.09491058517119825774728550794, −6.87648979877738163739533862848, −5.65881366875758050178889687322, −4.62812917836719298244163431833, −3.75174280427556281061563316708, −2.79082784679866247875160570608, −2.54044496831589407334097789043, −1.64228023370119954292412843957, 0, 1.64228023370119954292412843957, 2.54044496831589407334097789043, 2.79082784679866247875160570608, 3.75174280427556281061563316708, 4.62812917836719298244163431833, 5.65881366875758050178889687322, 6.87648979877738163739533862848, 7.09491058517119825774728550794, 7.947439369055605359251871590754

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