Properties

Label 2-43e2-1.1-c3-0-429
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  + 3.68·2-s + 8.25·3-s + 5.56·4-s + 12.2·5-s + 30.4·6-s − 28.5·7-s − 8.95·8-s + 41.1·9-s + 45.2·10-s − 55.8·11-s + 45.9·12-s − 70.9·13-s − 105.·14-s + 101.·15-s − 77.5·16-s + 20.9·17-s + 151.·18-s − 25.4·19-s + 68.4·20-s − 235.·21-s − 205.·22-s + 103.·23-s − 73.9·24-s + 26.1·25-s − 261.·26-s + 116.·27-s − 159.·28-s + ⋯
L(s)  = 1  + 1.30·2-s + 1.58·3-s + 0.696·4-s + 1.09·5-s + 2.06·6-s − 1.54·7-s − 0.395·8-s + 1.52·9-s + 1.43·10-s − 1.53·11-s + 1.10·12-s − 1.51·13-s − 2.00·14-s + 1.74·15-s − 1.21·16-s + 0.298·17-s + 1.98·18-s − 0.307·19-s + 0.765·20-s − 2.44·21-s − 1.99·22-s + 0.940·23-s − 0.628·24-s + 0.208·25-s − 1.97·26-s + 0.832·27-s − 1.07·28-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 \( 1 - 3.68T + 8T^{2} \)
3 \( 1 - 8.25T + 27T^{2} \)
5 \( 1 - 12.2T + 125T^{2} \)
7 \( 1 + 28.5T + 343T^{2} \)
11 \( 1 + 55.8T + 1.33e3T^{2} \)
13 \( 1 + 70.9T + 2.19e3T^{2} \)
17 \( 1 - 20.9T + 4.91e3T^{2} \)
19 \( 1 + 25.4T + 6.85e3T^{2} \)
23 \( 1 - 103.T + 1.21e4T^{2} \)
29 \( 1 + 94.0T + 2.43e4T^{2} \)
31 \( 1 - 16.0T + 2.97e4T^{2} \)
37 \( 1 - 23.8T + 5.06e4T^{2} \)
41 \( 1 + 75.5T + 6.89e4T^{2} \)
47 \( 1 + 5.56T + 1.03e5T^{2} \)
53 \( 1 - 622.T + 1.48e5T^{2} \)
59 \( 1 - 189.T + 2.05e5T^{2} \)
61 \( 1 - 371.T + 2.26e5T^{2} \)
67 \( 1 + 620.T + 3.00e5T^{2} \)
71 \( 1 + 644.T + 3.57e5T^{2} \)
73 \( 1 + 488.T + 3.89e5T^{2} \)
79 \( 1 + 1.11e3T + 4.93e5T^{2} \)
83 \( 1 + 351.T + 5.71e5T^{2} \)
89 \( 1 - 429.T + 7.04e5T^{2} \)
97 \( 1 - 1.18e3T + 9.12e5T^{2} \)
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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

−8.669362575978401403996168775279, −7.48176037068753450461314919730, −6.91378134290462835599555902845, −5.83147444158751344367389209916, −5.24828694399488851018245399143, −4.21307023125792269034456257524, −3.13456106103729142372219182901, −2.75072981210770160087600619160, −2.16611820810028583773082414421, 0, 2.16611820810028583773082414421, 2.75072981210770160087600619160, 3.13456106103729142372219182901, 4.21307023125792269034456257524, 5.24828694399488851018245399143, 5.83147444158751344367389209916, 6.91378134290462835599555902845, 7.48176037068753450461314919730, 8.669362575978401403996168775279

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