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 − 3.03·3-s + 4-s + 2.76·5-s + 3.03·6-s − 0.589·7-s − 8-s + 6.22·9-s − 2.76·10-s − 4.13·11-s − 3.03·12-s + 2.56·13-s + 0.589·14-s − 8.41·15-s + 16-s + 1.30·17-s − 6.22·18-s − 5.13·19-s + 2.76·20-s + 1.79·21-s + 4.13·22-s − 23-s + 3.03·24-s + 2.66·25-s − 2.56·26-s − 9.81·27-s − 0.589·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.75·3-s + 0.5·4-s + 1.23·5-s + 1.24·6-s − 0.222·7-s − 0.353·8-s + 2.07·9-s − 0.875·10-s − 1.24·11-s − 0.877·12-s + 0.710·13-s + 0.157·14-s − 2.17·15-s + 0.250·16-s + 0.316·17-s − 1.46·18-s − 1.17·19-s + 0.619·20-s + 0.391·21-s + 0.880·22-s − 0.208·23-s + 0.620·24-s + 0.533·25-s − 0.502·26-s − 1.88·27-s − 0.111·28-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 \( 1 + 3.03T + 3T^{2} \)
5 \( 1 - 2.76T + 5T^{2} \)
7 \( 1 + 0.589T + 7T^{2} \)
11 \( 1 + 4.13T + 11T^{2} \)
13 \( 1 - 2.56T + 13T^{2} \)
17 \( 1 - 1.30T + 17T^{2} \)
19 \( 1 + 5.13T + 19T^{2} \)
29 \( 1 - 3.77T + 29T^{2} \)
31 \( 1 - 6.58T + 31T^{2} \)
37 \( 1 - 6.66T + 37T^{2} \)
41 \( 1 + 0.317T + 41T^{2} \)
43 \( 1 + 4.28T + 43T^{2} \)
47 \( 1 - 6.67T + 47T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 + 6.57T + 59T^{2} \)
61 \( 1 + 4.26T + 61T^{2} \)
67 \( 1 + 6.71T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 5.20T + 73T^{2} \)
79 \( 1 + 4.85T + 79T^{2} \)
83 \( 1 - 5.44T + 83T^{2} \)
89 \( 1 + 4.24T + 89T^{2} \)
97 \( 1 + 9.63T + 97T^{2} \)
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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

−7.74052915150199461043631880011, −6.64130780969692271698492084081, −6.27913230667847549750577381229, −5.84314252603430440149868188913, −5.08025398836871205765148464703, −4.39998934627278756666476591059, −2.96076412139964254870445410941, −1.98484860507509390748366056506, −1.08180691042523923952763789630, 0, 1.08180691042523923952763789630, 1.98484860507509390748366056506, 2.96076412139964254870445410941, 4.39998934627278756666476591059, 5.08025398836871205765148464703, 5.84314252603430440149868188913, 6.27913230667847549750577381229, 6.64130780969692271698492084081, 7.74052915150199461043631880011

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