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 + 1.71·5-s − 1.11·6-s + 3.78·7-s − 8-s − 1.74·9-s − 1.71·10-s − 5.02·11-s + 1.11·12-s − 1.13·13-s − 3.78·14-s + 1.92·15-s + 16-s − 2.02·17-s + 1.74·18-s − 3.45·19-s + 1.71·20-s + 4.23·21-s + 5.02·22-s − 23-s − 1.11·24-s − 2.04·25-s + 1.13·26-s − 5.31·27-s + 3.78·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.646·3-s + 0.5·4-s + 0.768·5-s − 0.457·6-s + 1.43·7-s − 0.353·8-s − 0.582·9-s − 0.543·10-s − 1.51·11-s + 0.323·12-s − 0.315·13-s − 1.01·14-s + 0.496·15-s + 0.250·16-s − 0.490·17-s + 0.411·18-s − 0.791·19-s + 0.384·20-s + 0.924·21-s + 1.07·22-s − 0.208·23-s − 0.228·24-s − 0.409·25-s + 0.223·26-s − 1.02·27-s + 0.715·28-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 \( 1 - 1.71T + 5T^{2} \)
7 \( 1 - 3.78T + 7T^{2} \)
11 \( 1 + 5.02T + 11T^{2} \)
13 \( 1 + 1.13T + 13T^{2} \)
17 \( 1 + 2.02T + 17T^{2} \)
19 \( 1 + 3.45T + 19T^{2} \)
29 \( 1 - 3.63T + 29T^{2} \)
31 \( 1 + 2.60T + 31T^{2} \)
37 \( 1 - 6.85T + 37T^{2} \)
41 \( 1 - 0.0708T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 9.72T + 47T^{2} \)
53 \( 1 - 4.37T + 53T^{2} \)
59 \( 1 + 4.09T + 59T^{2} \)
61 \( 1 + 7.58T + 61T^{2} \)
67 \( 1 - 6.77T + 67T^{2} \)
71 \( 1 - 7.67T + 71T^{2} \)
73 \( 1 - 0.694T + 73T^{2} \)
79 \( 1 + 6.56T + 79T^{2} \)
83 \( 1 + 6.10T + 83T^{2} \)
89 \( 1 - 0.281T + 89T^{2} \)
97 \( 1 + 5.35T + 97T^{2} \)
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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

−8.043021333901518475109113668779, −7.34119960548620037228970642398, −6.33396752548712466032847451551, −5.53591055479214729368150393201, −5.01412584900871840440568492393, −4.03424756790414605180786215542, −2.64806607556856432721459735348, −2.38210077868395471805531756399, −1.53752583765057390059839584958, 0, 1.53752583765057390059839584958, 2.38210077868395471805531756399, 2.64806607556856432721459735348, 4.03424756790414605180786215542, 5.01412584900871840440568492393, 5.53591055479214729368150393201, 6.33396752548712466032847451551, 7.34119960548620037228970642398, 8.043021333901518475109113668779

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