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 + 1.72·3-s + 4-s + 0.546·5-s − 1.72·6-s + 2.37·7-s − 8-s − 0.0170·9-s − 0.546·10-s − 1.95·11-s + 1.72·12-s − 3.17·13-s − 2.37·14-s + 0.943·15-s + 16-s − 0.643·17-s + 0.0170·18-s + 0.483·19-s + 0.546·20-s + 4.11·21-s + 1.95·22-s − 23-s − 1.72·24-s − 4.70·25-s + 3.17·26-s − 5.21·27-s + 2.37·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.997·3-s + 0.5·4-s + 0.244·5-s − 0.705·6-s + 0.899·7-s − 0.353·8-s − 0.00567·9-s − 0.172·10-s − 0.589·11-s + 0.498·12-s − 0.880·13-s − 0.636·14-s + 0.243·15-s + 0.250·16-s − 0.155·17-s + 0.00401·18-s + 0.110·19-s + 0.122·20-s + 0.896·21-s + 0.416·22-s − 0.208·23-s − 0.352·24-s − 0.940·25-s + 0.622·26-s − 1.00·27-s + 0.449·28-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 \( 1 - 1.72T + 3T^{2} \)
5 \( 1 - 0.546T + 5T^{2} \)
7 \( 1 - 2.37T + 7T^{2} \)
11 \( 1 + 1.95T + 11T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 + 0.643T + 17T^{2} \)
19 \( 1 - 0.483T + 19T^{2} \)
29 \( 1 - 6.04T + 29T^{2} \)
31 \( 1 + 4.23T + 31T^{2} \)
37 \( 1 + 6.91T + 37T^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 - 7.54T + 43T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + 3.90T + 59T^{2} \)
61 \( 1 - 1.06T + 61T^{2} \)
67 \( 1 + 9.73T + 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 + 4.64T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + 5.56T + 89T^{2} \)
97 \( 1 - 8.21T + 97T^{2} \)
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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

−7.71241286200303897892594541856, −7.51727590103158553366244557867, −6.46753203607473443087879284831, −5.55029801011522306670932585382, −4.89439022192701498625067075507, −3.89183838912758223968561650937, −2.89734808145265020696444581968, −2.27882799826164816639175566307, −1.54940192013288765003172012897, 0, 1.54940192013288765003172012897, 2.27882799826164816639175566307, 2.89734808145265020696444581968, 3.89183838912758223968561650937, 4.89439022192701498625067075507, 5.55029801011522306670932585382, 6.46753203607473443087879284831, 7.51727590103158553366244557867, 7.71241286200303897892594541856

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