Properties

Label 2-43e2-1.1-c3-0-179
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  + 0.188·2-s + 1.07·3-s − 7.96·4-s − 16.0·5-s + 0.202·6-s − 29.7·7-s − 3.01·8-s − 25.8·9-s − 3.03·10-s + 12.2·11-s − 8.54·12-s + 37.3·13-s − 5.61·14-s − 17.2·15-s + 63.1·16-s + 35.5·17-s − 4.87·18-s − 95.8·19-s + 127.·20-s − 31.9·21-s + 2.31·22-s + 123.·23-s − 3.23·24-s + 132.·25-s + 7.04·26-s − 56.7·27-s + 236.·28-s + ⋯
L(s)  = 1  + 0.0667·2-s + 0.206·3-s − 0.995·4-s − 1.43·5-s + 0.0137·6-s − 1.60·7-s − 0.133·8-s − 0.957·9-s − 0.0958·10-s + 0.336·11-s − 0.205·12-s + 0.795·13-s − 0.107·14-s − 0.296·15-s + 0.986·16-s + 0.506·17-s − 0.0638·18-s − 1.15·19-s + 1.43·20-s − 0.331·21-s + 0.0224·22-s + 1.11·23-s − 0.0274·24-s + 1.06·25-s + 0.0531·26-s − 0.404·27-s + 1.59·28-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 \( 1 - 0.188T + 8T^{2} \)
3 \( 1 - 1.07T + 27T^{2} \)
5 \( 1 + 16.0T + 125T^{2} \)
7 \( 1 + 29.7T + 343T^{2} \)
11 \( 1 - 12.2T + 1.33e3T^{2} \)
13 \( 1 - 37.3T + 2.19e3T^{2} \)
17 \( 1 - 35.5T + 4.91e3T^{2} \)
19 \( 1 + 95.8T + 6.85e3T^{2} \)
23 \( 1 - 123.T + 1.21e4T^{2} \)
29 \( 1 - 194.T + 2.43e4T^{2} \)
31 \( 1 + 64.0T + 2.97e4T^{2} \)
37 \( 1 + 286.T + 5.06e4T^{2} \)
41 \( 1 - 442.T + 6.89e4T^{2} \)
47 \( 1 + 463.T + 1.03e5T^{2} \)
53 \( 1 + 142.T + 1.48e5T^{2} \)
59 \( 1 - 496.T + 2.05e5T^{2} \)
61 \( 1 + 144.T + 2.26e5T^{2} \)
67 \( 1 - 713.T + 3.00e5T^{2} \)
71 \( 1 - 548.T + 3.57e5T^{2} \)
73 \( 1 - 514.T + 3.89e5T^{2} \)
79 \( 1 + 1.07e3T + 4.93e5T^{2} \)
83 \( 1 - 126.T + 5.71e5T^{2} \)
89 \( 1 + 77.2T + 7.04e5T^{2} \)
97 \( 1 + 200.T + 9.12e5T^{2} \)
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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

−8.578281469740668445337176406596, −7.961978829164334292171816330042, −6.84786436410220303237279715535, −6.17474852049520743239210302653, −5.14815328451076746776385762533, −4.05117145854473634469935213038, −3.54105229424790992789004093562, −2.89890718925799726197271176088, −0.78184415559056232058781649690, 0, 0.78184415559056232058781649690, 2.89890718925799726197271176088, 3.54105229424790992789004093562, 4.05117145854473634469935213038, 5.14815328451076746776385762533, 6.17474852049520743239210302653, 6.84786436410220303237279715535, 7.961978829164334292171816330042, 8.578281469740668445337176406596

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