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 − 1.97·3-s + 4-s − 3.11·5-s + 1.97·6-s − 3.94·7-s − 8-s + 0.909·9-s + 3.11·10-s − 4.52·11-s − 1.97·12-s − 5.66·13-s + 3.94·14-s + 6.16·15-s + 16-s + 1.59·17-s − 0.909·18-s − 6.00·19-s − 3.11·20-s + 7.79·21-s + 4.52·22-s + 23-s + 1.97·24-s + 4.71·25-s + 5.66·26-s + 4.13·27-s − 3.94·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.14·3-s + 0.5·4-s − 1.39·5-s + 0.807·6-s − 1.49·7-s − 0.353·8-s + 0.303·9-s + 0.985·10-s − 1.36·11-s − 0.570·12-s − 1.57·13-s + 1.05·14-s + 1.59·15-s + 0.250·16-s + 0.386·17-s − 0.214·18-s − 1.37·19-s − 0.696·20-s + 1.70·21-s + 0.964·22-s + 0.208·23-s + 0.403·24-s + 0.942·25-s + 1.11·26-s + 0.795·27-s − 0.745·28-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 \( 1 + 1.97T + 3T^{2} \)
5 \( 1 + 3.11T + 5T^{2} \)
7 \( 1 + 3.94T + 7T^{2} \)
11 \( 1 + 4.52T + 11T^{2} \)
13 \( 1 + 5.66T + 13T^{2} \)
17 \( 1 - 1.59T + 17T^{2} \)
19 \( 1 + 6.00T + 19T^{2} \)
29 \( 1 + 5.05T + 29T^{2} \)
31 \( 1 - 3.72T + 31T^{2} \)
37 \( 1 - 3.72T + 37T^{2} \)
41 \( 1 - 2.67T + 41T^{2} \)
43 \( 1 - 6.70T + 43T^{2} \)
47 \( 1 + 2.96T + 47T^{2} \)
53 \( 1 + 2.04T + 53T^{2} \)
59 \( 1 - 1.88T + 59T^{2} \)
61 \( 1 - 2.55T + 61T^{2} \)
67 \( 1 - 4.61T + 67T^{2} \)
71 \( 1 - 5.59T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 9.80T + 83T^{2} \)
89 \( 1 + 0.355T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−7.56136429929424482212172067202, −7.19044524745309832956818213793, −6.37787263997042374985095215259, −5.73902230440049315361806823105, −4.89420527043196454694714407779, −4.11884589395749143249985460794, −3.04654943877487540651907448684, −2.46259647805388547266498770300, −0.50940447870168803271480564601, 0, 0.50940447870168803271480564601, 2.46259647805388547266498770300, 3.04654943877487540651907448684, 4.11884589395749143249985460794, 4.89420527043196454694714407779, 5.73902230440049315361806823105, 6.37787263997042374985095215259, 7.19044524745309832956818213793, 7.56136429929424482212172067202

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