Properties

Degree $2$
Conductor $1849$
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

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1849} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1849,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.652267835\)
\(L(\frac12)\) \(\approx\) \(6.652267835\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.85404139070977, −19.18477277237202, −18.12668031472648, −17.54645531608820, −17.02944254694969, −16.04301640379084, −15.11096384221606, −14.57019500089705, −14.12756008959990, −13.80620851625847, −13.14675457413599, −12.49106921663726, −11.79355067123376, −10.73008158668190, −9.690193603225647, −9.351106314858612, −8.706626615531981, −7.462385986845079, −6.535873146898229, −5.988911010103601, −5.056229230917805, −4.382425410460997, −3.160240042539536, −2.554825397154321, −1.800475179792662, 1.800475179792662, 2.554825397154321, 3.160240042539536, 4.382425410460997, 5.056229230917805, 5.988911010103601, 6.535873146898229, 7.462385986845079, 8.706626615531981, 9.351106314858612, 9.690193603225647, 10.73008158668190, 11.79355067123376, 12.49106921663726, 13.14675457413599, 13.80620851625847, 14.12756008959990, 14.57019500089705, 15.11096384221606, 16.04301640379084, 17.02944254694969, 17.54645531608820, 18.12668031472648, 19.18477277237202, 19.85404139070977

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