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 − 1.90·3-s + 4-s − 1.40·5-s + 1.90·6-s − 1.81·7-s − 8-s + 0.624·9-s + 1.40·10-s + 1.73·11-s − 1.90·12-s − 2.64·13-s + 1.81·14-s + 2.67·15-s + 16-s + 1.78·17-s − 0.624·18-s − 2.37·19-s − 1.40·20-s + 3.45·21-s − 1.73·22-s − 23-s + 1.90·24-s − 3.03·25-s + 2.64·26-s + 4.52·27-s − 1.81·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.09·3-s + 0.5·4-s − 0.627·5-s + 0.777·6-s − 0.686·7-s − 0.353·8-s + 0.208·9-s + 0.443·10-s + 0.523·11-s − 0.549·12-s − 0.732·13-s + 0.485·14-s + 0.689·15-s + 0.250·16-s + 0.432·17-s − 0.147·18-s − 0.545·19-s − 0.313·20-s + 0.754·21-s − 0.370·22-s − 0.208·23-s + 0.388·24-s − 0.606·25-s + 0.518·26-s + 0.870·27-s − 0.343·28-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 \( 1 + 1.90T + 3T^{2} \)
5 \( 1 + 1.40T + 5T^{2} \)
7 \( 1 + 1.81T + 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
29 \( 1 - 0.0835T + 29T^{2} \)
31 \( 1 + 0.944T + 31T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 + 8.01T + 41T^{2} \)
43 \( 1 - 8.50T + 43T^{2} \)
47 \( 1 + 2.06T + 47T^{2} \)
53 \( 1 + 2.56T + 53T^{2} \)
59 \( 1 - 0.0652T + 59T^{2} \)
61 \( 1 - 4.73T + 61T^{2} \)
67 \( 1 + 5.83T + 67T^{2} \)
71 \( 1 - 2.32T + 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 8.91T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 9.35T + 97T^{2} \)
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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

−7.79068828756058209737897483613, −6.91922033753621651530243668733, −6.38933122192595594776684358656, −5.79949143474706414870433813424, −4.93014156185797726675895872991, −4.07097831983168994400526039031, −3.20328813508364001012641708746, −2.18136598902064919066698814907, −0.851344867339571144973938938290, 0, 0.851344867339571144973938938290, 2.18136598902064919066698814907, 3.20328813508364001012641708746, 4.07097831983168994400526039031, 4.93014156185797726675895872991, 5.79949143474706414870433813424, 6.38933122192595594776684358656, 6.91922033753621651530243668733, 7.79068828756058209737897483613

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