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 + 2.61·3-s + 4-s − 2.46·5-s − 2.61·6-s + 4.06·7-s − 8-s + 3.82·9-s + 2.46·10-s + 1.03·11-s + 2.61·12-s − 4.93·13-s − 4.06·14-s − 6.44·15-s + 16-s + 0.256·17-s − 3.82·18-s − 5.12·19-s − 2.46·20-s + 10.6·21-s − 1.03·22-s − 23-s − 2.61·24-s + 1.07·25-s + 4.93·26-s + 2.16·27-s + 4.06·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.50·3-s + 0.5·4-s − 1.10·5-s − 1.06·6-s + 1.53·7-s − 0.353·8-s + 1.27·9-s + 0.779·10-s + 0.310·11-s + 0.754·12-s − 1.36·13-s − 1.08·14-s − 1.66·15-s + 0.250·16-s + 0.0623·17-s − 0.902·18-s − 1.17·19-s − 0.551·20-s + 2.31·21-s − 0.219·22-s − 0.208·23-s − 0.533·24-s + 0.214·25-s + 0.967·26-s + 0.416·27-s + 0.767·28-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 \( 1 - 2.61T + 3T^{2} \)
5 \( 1 + 2.46T + 5T^{2} \)
7 \( 1 - 4.06T + 7T^{2} \)
11 \( 1 - 1.03T + 11T^{2} \)
13 \( 1 + 4.93T + 13T^{2} \)
17 \( 1 - 0.256T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
29 \( 1 + 3.96T + 29T^{2} \)
31 \( 1 + 0.0806T + 31T^{2} \)
37 \( 1 + 4.61T + 37T^{2} \)
41 \( 1 + 4.16T + 41T^{2} \)
43 \( 1 + 5.83T + 43T^{2} \)
47 \( 1 - 2.83T + 47T^{2} \)
53 \( 1 - 5.76T + 53T^{2} \)
59 \( 1 - 1.27T + 59T^{2} \)
61 \( 1 + 7.85T + 61T^{2} \)
67 \( 1 + 5.27T + 67T^{2} \)
71 \( 1 - 1.38T + 71T^{2} \)
73 \( 1 + 2.78T + 73T^{2} \)
79 \( 1 + 5.36T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 - 6.38T + 89T^{2} \)
97 \( 1 + 7.88T + 97T^{2} \)
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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

−7.79342719533994728533154271056, −7.48155619349967025507525392091, −6.82651490489515052630921420934, −5.46461641192417955352123862683, −4.50992566668611760628924732544, −4.01606665258633625839428955728, −3.07799574750335938295128165583, −2.16973232602760183981720825772, −1.61499123369325123840504706249, 0, 1.61499123369325123840504706249, 2.16973232602760183981720825772, 3.07799574750335938295128165583, 4.01606665258633625839428955728, 4.50992566668611760628924732544, 5.46461641192417955352123862683, 6.82651490489515052630921420934, 7.48155619349967025507525392091, 7.79342719533994728533154271056

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