Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 9·9-s − 21·11-s − 17·13-s + 16·16-s − 9·17-s + 3·23-s + 25·25-s + 19·31-s + 36·36-s + 39·41-s − 43·43-s − 84·44-s − 78·47-s + 49·49-s − 68·52-s + 63·53-s − 54·59-s + 64·64-s + 91·67-s − 36·68-s − 14·79-s + 81·81-s + 123·83-s + 12·92-s − 193·97-s − 189·99-s + ⋯
L(s)  = 1  + 4-s + 9-s − 1.90·11-s − 1.30·13-s + 16-s − 0.529·17-s + 3/23·23-s + 25-s + 0.612·31-s + 36-s + 0.951·41-s − 43-s − 1.90·44-s − 1.65·47-s + 49-s − 1.30·52-s + 1.18·53-s − 0.915·59-s + 64-s + 1.35·67-s − 0.529·68-s − 0.177·79-s + 81-s + 1.48·83-s + 3/23·92-s − 1.98·97-s − 1.90·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.23364\)
\(L(\frac12)\)  \(\approx\)  \(1.23364\)
\(L(2)\)   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 \( 1 + p T \)
good2 \( ( 1 - p T )( 1 + p T ) \)
3 \( ( 1 - p T )( 1 + p T ) \)
5 \( ( 1 - p T )( 1 + p T ) \)
7 \( ( 1 - p T )( 1 + p T ) \)
11 \( 1 + 21 T + p^{2} T^{2} \)
13 \( 1 + 17 T + p^{2} T^{2} \)
17 \( 1 + 9 T + p^{2} T^{2} \)
19 \( ( 1 - p T )( 1 + p T ) \)
23 \( 1 - 3 T + p^{2} T^{2} \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 - 19 T + p^{2} T^{2} \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( 1 - 39 T + p^{2} T^{2} \)
47 \( 1 + 78 T + p^{2} T^{2} \)
53 \( 1 - 63 T + p^{2} T^{2} \)
59 \( 1 + 54 T + p^{2} T^{2} \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( 1 - 91 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( ( 1 - p T )( 1 + p T ) \)
79 \( 1 + 14 T + p^{2} T^{2} \)
83 \( 1 - 123 T + p^{2} T^{2} \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 + 193 T + p^{2} T^{2} \)
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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

−15.68745798247645466816065074550, −14.93830790529910498444437549418, −13.18202833767239569934644068348, −12.29094373488513376318635980589, −10.81120730186677110004926208563, −9.963233779979263944915015630361, −7.888827227322648391797264880663, −6.89295534312803505021028515568, −5.03933818808426973842888393284, −2.54220056186190216196915798797, 2.54220056186190216196915798797, 5.03933818808426973842888393284, 6.89295534312803505021028515568, 7.888827227322648391797264880663, 9.963233779979263944915015630361, 10.81120730186677110004926208563, 12.29094373488513376318635980589, 13.18202833767239569934644068348, 14.93830790529910498444437549418, 15.68745798247645466816065074550

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