Properties

Degree $2$
Conductor $1849$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.20·2-s + 4.82·3-s + 9.69·4-s − 7.01·5-s + 20.3·6-s + 27.8·7-s + 7.14·8-s − 3.68·9-s − 29.4·10-s − 14.8·11-s + 46.8·12-s − 60.1·13-s + 117.·14-s − 33.8·15-s − 47.5·16-s − 53.5·17-s − 15.5·18-s − 77.3·19-s − 67.9·20-s + 134.·21-s − 62.5·22-s + 61.7·23-s + 34.4·24-s − 75.8·25-s − 253.·26-s − 148.·27-s + 270.·28-s + ⋯
L(s)  = 1  + 1.48·2-s + 0.929·3-s + 1.21·4-s − 0.627·5-s + 1.38·6-s + 1.50·7-s + 0.315·8-s − 0.136·9-s − 0.932·10-s − 0.407·11-s + 1.12·12-s − 1.28·13-s + 2.23·14-s − 0.582·15-s − 0.742·16-s − 0.764·17-s − 0.203·18-s − 0.934·19-s − 0.760·20-s + 1.39·21-s − 0.605·22-s + 0.559·23-s + 0.293·24-s − 0.606·25-s − 1.90·26-s − 1.05·27-s + 1.82·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-1$
Motivic weight: \(3\)
Character: $\chi_{1849} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1849,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.20T + 8T^{2} \)
3 \( 1 - 4.82T + 27T^{2} \)
5 \( 1 + 7.01T + 125T^{2} \)
7 \( 1 - 27.8T + 343T^{2} \)
11 \( 1 + 14.8T + 1.33e3T^{2} \)
13 \( 1 + 60.1T + 2.19e3T^{2} \)
17 \( 1 + 53.5T + 4.91e3T^{2} \)
19 \( 1 + 77.3T + 6.85e3T^{2} \)
23 \( 1 - 61.7T + 1.21e4T^{2} \)
29 \( 1 - 53.9T + 2.43e4T^{2} \)
31 \( 1 + 191.T + 2.97e4T^{2} \)
37 \( 1 + 366.T + 5.06e4T^{2} \)
41 \( 1 - 5.32T + 6.89e4T^{2} \)
47 \( 1 - 559.T + 1.03e5T^{2} \)
53 \( 1 - 97.6T + 1.48e5T^{2} \)
59 \( 1 - 873.T + 2.05e5T^{2} \)
61 \( 1 - 723.T + 2.26e5T^{2} \)
67 \( 1 + 982.T + 3.00e5T^{2} \)
71 \( 1 - 417.T + 3.57e5T^{2} \)
73 \( 1 - 219.T + 3.89e5T^{2} \)
79 \( 1 + 801.T + 4.93e5T^{2} \)
83 \( 1 + 792.T + 5.71e5T^{2} \)
89 \( 1 + 1.20e3T + 7.04e5T^{2} \)
97 \( 1 - 577.T + 9.12e5T^{2} \)
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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

−8.565631725668676533597401123050, −7.51740803028605300111157663233, −7.07593760393084854853808932541, −5.68501081882393987726684533928, −5.03314152230021632907269819588, −4.32651488899726102935789379105, −3.64188946681911267404262357641, −2.49122193116717377843265035160, −2.05496858607950194464815002499, 0, 2.05496858607950194464815002499, 2.49122193116717377843265035160, 3.64188946681911267404262357641, 4.32651488899726102935789379105, 5.03314152230021632907269819588, 5.68501081882393987726684533928, 7.07593760393084854853808932541, 7.51740803028605300111157663233, 8.565631725668676533597401123050

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