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 − 1.70·3-s + 4-s − 2.14·5-s + 1.70·6-s + 0.462·7-s − 8-s − 0.0885·9-s + 2.14·10-s − 3.41·11-s − 1.70·12-s − 6.04·13-s − 0.462·14-s + 3.66·15-s + 16-s − 2.77·17-s + 0.0885·18-s + 4.36·19-s − 2.14·20-s − 0.788·21-s + 3.41·22-s + 23-s + 1.70·24-s − 0.384·25-s + 6.04·26-s + 5.26·27-s + 0.462·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.985·3-s + 0.5·4-s − 0.960·5-s + 0.696·6-s + 0.174·7-s − 0.353·8-s − 0.0295·9-s + 0.679·10-s − 1.02·11-s − 0.492·12-s − 1.67·13-s − 0.123·14-s + 0.946·15-s + 0.250·16-s − 0.672·17-s + 0.0208·18-s + 1.00·19-s − 0.480·20-s − 0.172·21-s + 0.727·22-s + 0.208·23-s + 0.348·24-s − 0.0768·25-s + 1.18·26-s + 1.01·27-s + 0.0873·28-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 \( 1 + 1.70T + 3T^{2} \)
5 \( 1 + 2.14T + 5T^{2} \)
7 \( 1 - 0.462T + 7T^{2} \)
11 \( 1 + 3.41T + 11T^{2} \)
13 \( 1 + 6.04T + 13T^{2} \)
17 \( 1 + 2.77T + 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
29 \( 1 - 6.57T + 29T^{2} \)
31 \( 1 - 5.44T + 31T^{2} \)
37 \( 1 + 6.84T + 37T^{2} \)
41 \( 1 - 1.53T + 41T^{2} \)
43 \( 1 + 2.73T + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + 5.46T + 59T^{2} \)
61 \( 1 + 6.36T + 61T^{2} \)
67 \( 1 - 8.91T + 67T^{2} \)
71 \( 1 - 4.61T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 - 5.62T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + 5.27T + 89T^{2} \)
97 \( 1 + 0.514T + 97T^{2} \)
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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

−7.85900553918447045691094219686, −7.03171846831486156518479507858, −6.57050688383065182203390663704, −5.35541518837011565659180763363, −5.10881068187339663960048937152, −4.21153209855504715739986629653, −2.97980825996065986081788185743, −2.35328642973828310463901600718, −0.795886524362455280404955689776, 0, 0.795886524362455280404955689776, 2.35328642973828310463901600718, 2.97980825996065986081788185743, 4.21153209855504715739986629653, 5.10881068187339663960048937152, 5.35541518837011565659180763363, 6.57050688383065182203390663704, 7.03171846831486156518479507858, 7.85900553918447045691094219686

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