Properties

Label 2-43e2-1.1-c3-0-366
Degree $2$
Conductor $1849$
Sign $-1$
Analytic cond. $109.094$
Root an. cond. $10.4448$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.72·2-s − 5.79·3-s + 5.84·4-s + 14.8·5-s − 21.5·6-s + 12.2·7-s − 8.00·8-s + 6.59·9-s + 55.1·10-s + 2.05·11-s − 33.8·12-s + 9.53·13-s + 45.4·14-s − 85.8·15-s − 76.5·16-s − 74.1·17-s + 24.5·18-s + 103.·19-s + 86.6·20-s − 70.7·21-s + 7.63·22-s − 146.·23-s + 46.4·24-s + 94.4·25-s + 35.4·26-s + 118.·27-s + 71.4·28-s + ⋯
L(s)  = 1  + 1.31·2-s − 1.11·3-s + 0.731·4-s + 1.32·5-s − 1.46·6-s + 0.659·7-s − 0.353·8-s + 0.244·9-s + 1.74·10-s + 0.0562·11-s − 0.815·12-s + 0.203·13-s + 0.867·14-s − 1.47·15-s − 1.19·16-s − 1.05·17-s + 0.321·18-s + 1.24·19-s + 0.968·20-s − 0.735·21-s + 0.0740·22-s − 1.33·23-s + 0.394·24-s + 0.755·25-s + 0.267·26-s + 0.843·27-s + 0.481·28-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: $-1$
Analytic conductor: \(109.094\)
Root analytic conductor: \(10.4448\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
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 \( 1 - 3.72T + 8T^{2} \)
3 \( 1 + 5.79T + 27T^{2} \)
5 \( 1 - 14.8T + 125T^{2} \)
7 \( 1 - 12.2T + 343T^{2} \)
11 \( 1 - 2.05T + 1.33e3T^{2} \)
13 \( 1 - 9.53T + 2.19e3T^{2} \)
17 \( 1 + 74.1T + 4.91e3T^{2} \)
19 \( 1 - 103.T + 6.85e3T^{2} \)
23 \( 1 + 146.T + 1.21e4T^{2} \)
29 \( 1 - 102.T + 2.43e4T^{2} \)
31 \( 1 + 308.T + 2.97e4T^{2} \)
37 \( 1 + 434.T + 5.06e4T^{2} \)
41 \( 1 - 183.T + 6.89e4T^{2} \)
47 \( 1 - 348.T + 1.03e5T^{2} \)
53 \( 1 + 199.T + 1.48e5T^{2} \)
59 \( 1 + 156.T + 2.05e5T^{2} \)
61 \( 1 - 261.T + 2.26e5T^{2} \)
67 \( 1 + 43.3T + 3.00e5T^{2} \)
71 \( 1 - 654.T + 3.57e5T^{2} \)
73 \( 1 + 868.T + 3.89e5T^{2} \)
79 \( 1 + 1.35e3T + 4.93e5T^{2} \)
83 \( 1 - 166.T + 5.71e5T^{2} \)
89 \( 1 - 1.18e3T + 7.04e5T^{2} \)
97 \( 1 + 398.T + 9.12e5T^{2} \)
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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

−8.633992101283145715359858942677, −7.26984576758851128646589980459, −6.40243061710370588009692291469, −5.81430579796041913641514059563, −5.32660468228708295571436843931, −4.71341422388477214871890365273, −3.67482810498803039691532719289, −2.45100137960011576371257606543, −1.53477867629583613224833222124, 0, 1.53477867629583613224833222124, 2.45100137960011576371257606543, 3.67482810498803039691532719289, 4.71341422388477214871890365273, 5.32660468228708295571436843931, 5.81430579796041913641514059563, 6.40243061710370588009692291469, 7.26984576758851128646589980459, 8.633992101283145715359858942677

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