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 + 1.50·3-s + 4-s − 2.72·5-s − 1.50·6-s − 2.36·7-s − 8-s − 0.734·9-s + 2.72·10-s + 3.29·11-s + 1.50·12-s − 0.0258·13-s + 2.36·14-s − 4.10·15-s + 16-s + 1.25·17-s + 0.734·18-s − 1.42·19-s − 2.72·20-s − 3.55·21-s − 3.29·22-s + 23-s − 1.50·24-s + 2.42·25-s + 0.0258·26-s − 5.62·27-s − 2.36·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.868·3-s + 0.5·4-s − 1.21·5-s − 0.614·6-s − 0.893·7-s − 0.353·8-s − 0.244·9-s + 0.861·10-s + 0.992·11-s + 0.434·12-s − 0.00718·13-s + 0.632·14-s − 1.05·15-s + 0.250·16-s + 0.304·17-s + 0.173·18-s − 0.327·19-s − 0.609·20-s − 0.776·21-s − 0.702·22-s + 0.208·23-s − 0.307·24-s + 0.485·25-s + 0.00507·26-s − 1.08·27-s − 0.446·28-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 \( 1 - 1.50T + 3T^{2} \)
5 \( 1 + 2.72T + 5T^{2} \)
7 \( 1 + 2.36T + 7T^{2} \)
11 \( 1 - 3.29T + 11T^{2} \)
13 \( 1 + 0.0258T + 13T^{2} \)
17 \( 1 - 1.25T + 17T^{2} \)
19 \( 1 + 1.42T + 19T^{2} \)
29 \( 1 - 1.18T + 29T^{2} \)
31 \( 1 - 5.97T + 31T^{2} \)
37 \( 1 - 6.53T + 37T^{2} \)
41 \( 1 - 1.11T + 41T^{2} \)
43 \( 1 - 3.03T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + 8.79T + 53T^{2} \)
59 \( 1 + 0.637T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - 4.76T + 67T^{2} \)
71 \( 1 + 7.69T + 71T^{2} \)
73 \( 1 + 6.63T + 73T^{2} \)
79 \( 1 - 7.08T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 2.00T + 97T^{2} \)
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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

−7.82415378257529382564008149830, −7.31176114789711373815654908554, −6.45302401976823426956736463605, −5.91011355664431032733415258794, −4.50585307974997526877587472474, −3.82500040625599956058597140588, −3.14931789792027318760802562324, −2.50454967292491822434559229863, −1.14982863448888390230547221877, 0, 1.14982863448888390230547221877, 2.50454967292491822434559229863, 3.14931789792027318760802562324, 3.82500040625599956058597140588, 4.50585307974997526877587472474, 5.91011355664431032733415258794, 6.45302401976823426956736463605, 7.31176114789711373815654908554, 7.82415378257529382564008149830

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