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 + 2.17·3-s + 4-s + 2.64·5-s − 2.17·6-s + 0.866·7-s − 8-s + 1.74·9-s − 2.64·10-s − 2.06·11-s + 2.17·12-s − 5.30·13-s − 0.866·14-s + 5.75·15-s + 16-s − 4.65·17-s − 1.74·18-s − 1.86·19-s + 2.64·20-s + 1.88·21-s + 2.06·22-s + 23-s − 2.17·24-s + 1.97·25-s + 5.30·26-s − 2.73·27-s + 0.866·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.25·3-s + 0.5·4-s + 1.18·5-s − 0.889·6-s + 0.327·7-s − 0.353·8-s + 0.581·9-s − 0.835·10-s − 0.622·11-s + 0.628·12-s − 1.47·13-s − 0.231·14-s + 1.48·15-s + 0.250·16-s − 1.12·17-s − 0.411·18-s − 0.426·19-s + 0.590·20-s + 0.411·21-s + 0.440·22-s + 0.208·23-s − 0.444·24-s + 0.395·25-s + 1.04·26-s − 0.526·27-s + 0.163·28-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 \( 1 - 2.17T + 3T^{2} \)
5 \( 1 - 2.64T + 5T^{2} \)
7 \( 1 - 0.866T + 7T^{2} \)
11 \( 1 + 2.06T + 11T^{2} \)
13 \( 1 + 5.30T + 13T^{2} \)
17 \( 1 + 4.65T + 17T^{2} \)
19 \( 1 + 1.86T + 19T^{2} \)
29 \( 1 + 5.76T + 29T^{2} \)
31 \( 1 + 1.84T + 31T^{2} \)
37 \( 1 + 9.12T + 37T^{2} \)
41 \( 1 + 3.84T + 41T^{2} \)
43 \( 1 - 8.82T + 43T^{2} \)
47 \( 1 - 7.58T + 47T^{2} \)
53 \( 1 + 7.00T + 53T^{2} \)
59 \( 1 - 4.34T + 59T^{2} \)
61 \( 1 - 0.363T + 61T^{2} \)
67 \( 1 - 0.0197T + 67T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + 0.613T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−7.80702807606142870385835826955, −7.27056237114590936170883050278, −6.53206201102045176204734403212, −5.56130914420686349835876524246, −4.95030615593044258491104140406, −3.85036075575387709349780009081, −2.76323163959399067585162048388, −2.22883364928201523490157705772, −1.77665827740375787441580608098, 0, 1.77665827740375787441580608098, 2.22883364928201523490157705772, 2.76323163959399067585162048388, 3.85036075575387709349780009081, 4.95030615593044258491104140406, 5.56130914420686349835876524246, 6.53206201102045176204734403212, 7.27056237114590936170883050278, 7.80702807606142870385835826955

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