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 + 1.11·3-s + 4-s + 0.636·5-s − 1.11·6-s + 1.86·7-s − 8-s − 1.74·9-s − 0.636·10-s − 3.79·11-s + 1.11·12-s + 4.96·13-s − 1.86·14-s + 0.712·15-s + 16-s + 5.06·17-s + 1.74·18-s − 4.24·19-s + 0.636·20-s + 2.08·21-s + 3.79·22-s + 23-s − 1.11·24-s − 4.59·25-s − 4.96·26-s − 5.31·27-s + 1.86·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.646·3-s + 0.5·4-s + 0.284·5-s − 0.457·6-s + 0.705·7-s − 0.353·8-s − 0.582·9-s − 0.201·10-s − 1.14·11-s + 0.323·12-s + 1.37·13-s − 0.498·14-s + 0.184·15-s + 0.250·16-s + 1.22·17-s + 0.411·18-s − 0.973·19-s + 0.142·20-s + 0.456·21-s + 0.809·22-s + 0.208·23-s − 0.228·24-s − 0.918·25-s − 0.974·26-s − 1.02·27-s + 0.352·28-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 \( 1 - 1.11T + 3T^{2} \)
5 \( 1 - 0.636T + 5T^{2} \)
7 \( 1 - 1.86T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 - 4.96T + 13T^{2} \)
17 \( 1 - 5.06T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
29 \( 1 + 4.06T + 29T^{2} \)
31 \( 1 + 0.357T + 31T^{2} \)
37 \( 1 + 7.63T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 + 4.52T + 43T^{2} \)
47 \( 1 - 5.70T + 47T^{2} \)
53 \( 1 + 2.51T + 53T^{2} \)
59 \( 1 - 2.98T + 59T^{2} \)
61 \( 1 - 0.0976T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 2.78T + 73T^{2} \)
79 \( 1 + 7.67T + 79T^{2} \)
83 \( 1 - 8.91T + 83T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 + 4.30T + 97T^{2} \)
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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

−7.911165496530734736815182889907, −7.36917372250512570148309241726, −6.27012406060066367016805654259, −5.67280045180310832100085804473, −5.01322415815682415977934799024, −3.71847309130705953040438025995, −3.15709664313818253436331786458, −2.12462234683672080057608473320, −1.50323182184173167699525152506, 0, 1.50323182184173167699525152506, 2.12462234683672080057608473320, 3.15709664313818253436331786458, 3.71847309130705953040438025995, 5.01322415815682415977934799024, 5.67280045180310832100085804473, 6.27012406060066367016805654259, 7.36917372250512570148309241726, 7.911165496530734736815182889907

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