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 + 0.977·3-s + 4-s + 1.34·5-s − 0.977·6-s − 1.08·7-s − 8-s − 2.04·9-s − 1.34·10-s + 3.05·11-s + 0.977·12-s + 1.13·13-s + 1.08·14-s + 1.31·15-s + 16-s − 3.38·17-s + 2.04·18-s + 2.02·19-s + 1.34·20-s − 1.06·21-s − 3.05·22-s − 23-s − 0.977·24-s − 3.19·25-s − 1.13·26-s − 4.93·27-s − 1.08·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.564·3-s + 0.5·4-s + 0.600·5-s − 0.399·6-s − 0.411·7-s − 0.353·8-s − 0.681·9-s − 0.424·10-s + 0.921·11-s + 0.282·12-s + 0.315·13-s + 0.290·14-s + 0.338·15-s + 0.250·16-s − 0.822·17-s + 0.481·18-s + 0.464·19-s + 0.300·20-s − 0.231·21-s − 0.651·22-s − 0.208·23-s − 0.199·24-s − 0.639·25-s − 0.223·26-s − 0.948·27-s − 0.205·28-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 \( 1 - 0.977T + 3T^{2} \)
5 \( 1 - 1.34T + 5T^{2} \)
7 \( 1 + 1.08T + 7T^{2} \)
11 \( 1 - 3.05T + 11T^{2} \)
13 \( 1 - 1.13T + 13T^{2} \)
17 \( 1 + 3.38T + 17T^{2} \)
19 \( 1 - 2.02T + 19T^{2} \)
29 \( 1 - 1.84T + 29T^{2} \)
31 \( 1 - 1.02T + 31T^{2} \)
37 \( 1 + 2.61T + 37T^{2} \)
41 \( 1 + 7.20T + 41T^{2} \)
43 \( 1 + 5.50T + 43T^{2} \)
47 \( 1 - 5.56T + 47T^{2} \)
53 \( 1 + 13.2T + 53T^{2} \)
59 \( 1 - 5.29T + 59T^{2} \)
61 \( 1 - 4.01T + 61T^{2} \)
67 \( 1 + 9.80T + 67T^{2} \)
71 \( 1 + 7.93T + 71T^{2} \)
73 \( 1 - 3.80T + 73T^{2} \)
79 \( 1 - 2.61T + 79T^{2} \)
83 \( 1 - 9.13T + 83T^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 - 1.60T + 97T^{2} \)
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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

−7.931228364409784157900886178712, −7.01385643395413946181976088491, −6.37292742264142054246598804962, −5.87444822391796666537741522960, −4.88128338755145989274425590255, −3.75522821627377241929319421947, −3.11503723197837660487220419969, −2.19973463582631219137419612504, −1.43540068721342923380809077680, 0, 1.43540068721342923380809077680, 2.19973463582631219137419612504, 3.11503723197837660487220419969, 3.75522821627377241929319421947, 4.88128338755145989274425590255, 5.87444822391796666537741522960, 6.37292742264142054246598804962, 7.01385643395413946181976088491, 7.931228364409784157900886178712

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