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 + 2.60·3-s + 4-s + 1.63·5-s − 2.60·6-s − 2.25·7-s − 8-s + 3.79·9-s − 1.63·10-s + 4.17·11-s + 2.60·12-s − 1.43·13-s + 2.25·14-s + 4.27·15-s + 16-s − 6.17·17-s − 3.79·18-s − 8.47·19-s + 1.63·20-s − 5.86·21-s − 4.17·22-s − 23-s − 2.60·24-s − 2.31·25-s + 1.43·26-s + 2.06·27-s − 2.25·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.50·3-s + 0.5·4-s + 0.732·5-s − 1.06·6-s − 0.851·7-s − 0.353·8-s + 1.26·9-s − 0.518·10-s + 1.25·11-s + 0.752·12-s − 0.396·13-s + 0.601·14-s + 1.10·15-s + 0.250·16-s − 1.49·17-s − 0.893·18-s − 1.94·19-s + 0.366·20-s − 1.28·21-s − 0.889·22-s − 0.208·23-s − 0.531·24-s − 0.463·25-s + 0.280·26-s + 0.397·27-s − 0.425·28-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 \( 1 - 2.60T + 3T^{2} \)
5 \( 1 - 1.63T + 5T^{2} \)
7 \( 1 + 2.25T + 7T^{2} \)
11 \( 1 - 4.17T + 11T^{2} \)
13 \( 1 + 1.43T + 13T^{2} \)
17 \( 1 + 6.17T + 17T^{2} \)
19 \( 1 + 8.47T + 19T^{2} \)
29 \( 1 + 6.96T + 29T^{2} \)
31 \( 1 - 0.614T + 31T^{2} \)
37 \( 1 - 3.72T + 37T^{2} \)
41 \( 1 + 0.876T + 41T^{2} \)
43 \( 1 - 3.75T + 43T^{2} \)
47 \( 1 + 2.27T + 47T^{2} \)
53 \( 1 + 3.45T + 53T^{2} \)
59 \( 1 - 1.67T + 59T^{2} \)
61 \( 1 + 14.9T + 61T^{2} \)
67 \( 1 - 2.88T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 + 3.68T + 73T^{2} \)
79 \( 1 + 2.58T + 79T^{2} \)
83 \( 1 - 3.75T + 83T^{2} \)
89 \( 1 - 7.45T + 89T^{2} \)
97 \( 1 - 2.64T + 97T^{2} \)
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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

−7.85256545644109039199193812766, −7.15006789554908370759743505398, −6.33078041028644845400202807043, −6.10969995192314700711962360799, −4.48599153731970198465873547807, −3.88949028774940541356661851386, −2.99443444592132004383373057104, −2.12637424714621166176423519756, −1.77680333421538399035881961388, 0, 1.77680333421538399035881961388, 2.12637424714621166176423519756, 2.99443444592132004383373057104, 3.88949028774940541356661851386, 4.48599153731970198465873547807, 6.10969995192314700711962360799, 6.33078041028644845400202807043, 7.15006789554908370759743505398, 7.85256545644109039199193812766

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