Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 2·4-s + 4·5-s + 4·6-s + 9-s + 8·10-s + 3·11-s + 4·12-s − 5·13-s + 8·15-s − 4·16-s − 3·17-s + 2·18-s + 2·19-s + 8·20-s + 6·22-s − 23-s + 11·25-s − 10·26-s − 4·27-s + 6·29-s + 16·30-s − 31-s − 8·32-s + 6·33-s − 6·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 4-s + 1.78·5-s + 1.63·6-s + 1/3·9-s + 2.52·10-s + 0.904·11-s + 1.15·12-s − 1.38·13-s + 2.06·15-s − 16-s − 0.727·17-s + 0.471·18-s + 0.458·19-s + 1.78·20-s + 1.27·22-s − 0.208·23-s + 11/5·25-s − 1.96·26-s − 0.769·27-s + 1.11·29-s + 2.92·30-s − 0.179·31-s − 1.41·32-s + 1.04·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1849\)    =    \(43^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1849} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1849,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(6.652267835\)
\(L(\frac12)\)  \(\approx\)  \(6.652267835\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−9.351106314858612004769456814696, −8.706626615531981381764341189714, −7.46238598684507898242070539770, −6.53587314689822873386512255529, −5.98891101010360117124996175140, −5.05622923091780464116124937114, −4.38242541046099734716547640882, −3.16024004253953616612721273066, −2.55482539715432140211864034960, −1.80047517979266187156874628712, 1.80047517979266187156874628712, 2.55482539715432140211864034960, 3.16024004253953616612721273066, 4.38242541046099734716547640882, 5.05622923091780464116124937114, 5.98891101010360117124996175140, 6.53587314689822873386512255529, 7.46238598684507898242070539770, 8.706626615531981381764341189714, 9.351106314858612004769456814696

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