Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 10
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.02e3·4-s + 5.90e4·9-s − 1.85e4·11-s + 3.03e5·13-s + 1.04e6·16-s − 2.76e6·17-s + 4.12e6·23-s + 9.76e6·25-s + 5.72e7·31-s + 6.04e7·36-s + 1.42e8·41-s − 1.47e8·43-s − 1.89e7·44-s + 4.51e8·47-s + 2.82e8·49-s + 3.11e8·52-s − 3.39e7·53-s − 9.90e8·59-s + 1.07e9·64-s − 1.50e9·67-s − 2.83e9·68-s − 2.64e9·79-s + 3.48e9·81-s − 6.75e9·83-s + 4.22e9·92-s − 1.50e10·97-s − 1.09e9·99-s + ⋯
L(s)  = 1  + 4-s + 9-s − 0.114·11-s + 0.818·13-s + 16-s − 1.94·17-s + 0.641·23-s + 25-s + 1.99·31-s + 36-s + 1.23·41-s − 43-s − 0.114·44-s + 1.96·47-s + 49-s + 0.818·52-s − 0.0812·53-s − 1.38·59-s + 64-s − 1.11·67-s − 1.94·68-s − 0.858·79-s + 81-s − 1.71·83-s + 0.641·92-s − 1.74·97-s − 0.114·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ 1)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(2.80969\)
\(L(\frac12)\)  \(\approx\)  \(2.80969\)
\(L(6)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 + p^{5} T \)
good2 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
3 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
5 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
7 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
11 \( 1 + 18501 T + p^{10} T^{2} \)
13 \( 1 - 303943 T + p^{10} T^{2} \)
17 \( 1 + 2764089 T + p^{10} T^{2} \)
19 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
23 \( 1 - 4126443 T + p^{10} T^{2} \)
29 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
31 \( 1 - 57253099 T + p^{10} T^{2} \)
37 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
41 \( 1 - 142671399 T + p^{10} T^{2} \)
47 \( 1 - 451176882 T + p^{10} T^{2} \)
53 \( 1 + 33972057 T + p^{10} T^{2} \)
59 \( 1 + 990191574 T + p^{10} T^{2} \)
61 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
67 \( 1 + 1504819589 T + p^{10} T^{2} \)
71 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
73 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
79 \( 1 + 2641416974 T + p^{10} T^{2} \)
83 \( 1 + 6757639557 T + p^{10} T^{2} \)
89 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
97 \( 1 + 15006753793 T + p^{10} T^{2} \)
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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

−13.61968985847640932962343317376, −12.55575504167204922346203540064, −11.23475373402581615493898358447, −10.38750795901089135860605357578, −8.766017487478642134785561873987, −7.22161228931604059319959815758, −6.28754246321329285837254271214, −4.39527417219450588265669809377, −2.62727087722162381857371365653, −1.17894872103903504229698025076, 1.17894872103903504229698025076, 2.62727087722162381857371365653, 4.39527417219450588265669809377, 6.28754246321329285837254271214, 7.22161228931604059319959815758, 8.766017487478642134785561873987, 10.38750795901089135860605357578, 11.23475373402581615493898358447, 12.55575504167204922346203540064, 13.61968985847640932962343317376

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