Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 8
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 256·4-s + 6.56e3·9-s + 1.03e4·11-s − 5.47e4·13-s + 6.55e4·16-s + 7.99e4·17-s + 5.40e5·23-s + 3.90e5·25-s + 5.89e5·31-s + 1.67e6·36-s − 2.26e6·41-s + 3.41e6·43-s + 2.64e6·44-s − 6.98e6·47-s + 5.76e6·49-s − 1.40e7·52-s − 1.30e7·53-s − 7.86e6·59-s + 1.67e7·64-s − 3.98e7·67-s + 2.04e7·68-s + 7.30e7·79-s + 4.30e7·81-s − 9.30e7·83-s + 1.38e8·92-s + 1.62e8·97-s + 6.77e7·99-s + ⋯
L(s)  = 1  + 4-s + 9-s + 0.704·11-s − 1.91·13-s + 16-s + 0.957·17-s + 1.93·23-s + 25-s + 0.638·31-s + 36-s − 0.800·41-s + 43-s + 0.704·44-s − 1.43·47-s + 49-s − 1.91·52-s − 1.65·53-s − 0.649·59-s + 64-s − 1.97·67-s + 0.957·68-s + 1.87·79-s + 81-s − 1.96·83-s + 1.93·92-s + 1.83·97-s + 0.704·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 1)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(2.58582\)
\(L(\frac12)\)  \(\approx\)  \(2.58582\)
\(L(5)\)   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 \( 1 - p^{4} T \)
good2 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
3 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
5 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
7 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
11 \( 1 - 10319 T + p^{8} T^{2} \)
13 \( 1 + 54721 T + p^{8} T^{2} \)
17 \( 1 - 79967 T + p^{8} T^{2} \)
19 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
23 \( 1 - 540719 T + p^{8} T^{2} \)
29 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
31 \( 1 - 589679 T + p^{8} T^{2} \)
37 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
41 \( 1 + 2262241 T + p^{8} T^{2} \)
47 \( 1 + 6983806 T + p^{8} T^{2} \)
53 \( 1 + 13061761 T + p^{8} T^{2} \)
59 \( 1 + 7864606 T + p^{8} T^{2} \)
61 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
67 \( 1 + 39816433 T + p^{8} T^{2} \)
71 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
73 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
79 \( 1 - 73045634 T + p^{8} T^{2} \)
83 \( 1 + 93091441 T + p^{8} T^{2} \)
89 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
97 \( 1 - 162643199 T + p^{8} T^{2} \)
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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

−14.53637338269310450238893522765, −12.72368526241077260352333713705, −11.95416634610319609975392492572, −10.57597117770374067667485560253, −9.496157751418395355863380369904, −7.54419779467933091952842877846, −6.73552743936133766429739537324, −4.89851293834514499211300123457, −2.93068206177018866790131117877, −1.29439260531378622269924005689, 1.29439260531378622269924005689, 2.93068206177018866790131117877, 4.89851293834514499211300123457, 6.73552743936133766429739537324, 7.54419779467933091952842877846, 9.496157751418395355863380369904, 10.57597117770374067667485560253, 11.95416634610319609975392492572, 12.72368526241077260352333713705, 14.53637338269310450238893522765

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