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  + 1.67·2-s + 3.03·3-s − 5.20·4-s − 19.6·5-s + 5.06·6-s + 28.4·7-s − 22.0·8-s − 17.8·9-s − 32.7·10-s − 28.1·11-s − 15.7·12-s + 20.8·13-s + 47.5·14-s − 59.4·15-s + 4.72·16-s + 24.2·17-s − 29.7·18-s + 118.·19-s + 102.·20-s + 86.2·21-s − 47.0·22-s + 61.3·23-s − 66.9·24-s + 259.·25-s + 34.8·26-s − 135.·27-s − 148.·28-s + ⋯
L(s)  = 1  + 0.591·2-s + 0.583·3-s − 0.650·4-s − 1.75·5-s + 0.344·6-s + 1.53·7-s − 0.975·8-s − 0.659·9-s − 1.03·10-s − 0.771·11-s − 0.379·12-s + 0.445·13-s + 0.908·14-s − 1.02·15-s + 0.0738·16-s + 0.345·17-s − 0.389·18-s + 1.43·19-s + 1.14·20-s + 0.896·21-s − 0.456·22-s + 0.556·23-s − 0.569·24-s + 2.07·25-s + 0.263·26-s − 0.968·27-s − 0.999·28-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 \( 1 - 1.67T + 8T^{2} \)
3 \( 1 - 3.03T + 27T^{2} \)
5 \( 1 + 19.6T + 125T^{2} \)
7 \( 1 - 28.4T + 343T^{2} \)
11 \( 1 + 28.1T + 1.33e3T^{2} \)
13 \( 1 - 20.8T + 2.19e3T^{2} \)
17 \( 1 - 24.2T + 4.91e3T^{2} \)
19 \( 1 - 118.T + 6.85e3T^{2} \)
23 \( 1 - 61.3T + 1.21e4T^{2} \)
29 \( 1 - 70.4T + 2.43e4T^{2} \)
31 \( 1 - 263.T + 2.97e4T^{2} \)
37 \( 1 + 360.T + 5.06e4T^{2} \)
41 \( 1 - 67.4T + 6.89e4T^{2} \)
47 \( 1 + 457.T + 1.03e5T^{2} \)
53 \( 1 + 148.T + 1.48e5T^{2} \)
59 \( 1 + 182.T + 2.05e5T^{2} \)
61 \( 1 - 215.T + 2.26e5T^{2} \)
67 \( 1 + 631.T + 3.00e5T^{2} \)
71 \( 1 + 575.T + 3.57e5T^{2} \)
73 \( 1 + 1.11e3T + 3.89e5T^{2} \)
79 \( 1 + 622.T + 4.93e5T^{2} \)
83 \( 1 + 475.T + 5.71e5T^{2} \)
89 \( 1 + 1.39e3T + 7.04e5T^{2} \)
97 \( 1 - 1.06e3T + 9.12e5T^{2} \)
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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

−8.462050570569153844565799452764, −7.88595951037619751428195699681, −7.27548281523983744891214921362, −5.74507355310813596448209849259, −4.90135197309170382439606709298, −4.48339715605392524808255239207, −3.37286325094905646175798895788, −2.96564873987302831245933854385, −1.17665603108512355394168117179, 0, 1.17665603108512355394168117179, 2.96564873987302831245933854385, 3.37286325094905646175798895788, 4.48339715605392524808255239207, 4.90135197309170382439606709298, 5.74507355310813596448209849259, 7.27548281523983744891214921362, 7.88595951037619751428195699681, 8.462050570569153844565799452764

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