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  − 2.06·2-s − 3.28·3-s − 3.72·4-s + 0.905·5-s + 6.80·6-s + 0.553·7-s + 24.2·8-s − 16.1·9-s − 1.87·10-s + 36.0·11-s + 12.2·12-s + 72.5·13-s − 1.14·14-s − 2.97·15-s − 20.3·16-s − 42.3·17-s + 33.4·18-s − 54.5·19-s − 3.37·20-s − 1.82·21-s − 74.5·22-s − 113.·23-s − 79.7·24-s − 124.·25-s − 149.·26-s + 142.·27-s − 2.06·28-s + ⋯
L(s)  = 1  − 0.731·2-s − 0.633·3-s − 0.465·4-s + 0.0809·5-s + 0.462·6-s + 0.0298·7-s + 1.07·8-s − 0.599·9-s − 0.0592·10-s + 0.988·11-s + 0.294·12-s + 1.54·13-s − 0.0218·14-s − 0.0512·15-s − 0.317·16-s − 0.604·17-s + 0.438·18-s − 0.658·19-s − 0.0377·20-s − 0.0189·21-s − 0.722·22-s − 1.03·23-s − 0.678·24-s − 0.993·25-s − 1.13·26-s + 1.01·27-s − 0.0139·28-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 \( 1 + 2.06T + 8T^{2} \)
3 \( 1 + 3.28T + 27T^{2} \)
5 \( 1 - 0.905T + 125T^{2} \)
7 \( 1 - 0.553T + 343T^{2} \)
11 \( 1 - 36.0T + 1.33e3T^{2} \)
13 \( 1 - 72.5T + 2.19e3T^{2} \)
17 \( 1 + 42.3T + 4.91e3T^{2} \)
19 \( 1 + 54.5T + 6.85e3T^{2} \)
23 \( 1 + 113.T + 1.21e4T^{2} \)
29 \( 1 + 18.9T + 2.43e4T^{2} \)
31 \( 1 - 45.5T + 2.97e4T^{2} \)
37 \( 1 - 56.1T + 5.06e4T^{2} \)
41 \( 1 + 51.8T + 6.89e4T^{2} \)
47 \( 1 - 133.T + 1.03e5T^{2} \)
53 \( 1 + 368.T + 1.48e5T^{2} \)
59 \( 1 - 489.T + 2.05e5T^{2} \)
61 \( 1 - 592.T + 2.26e5T^{2} \)
67 \( 1 - 1.06e3T + 3.00e5T^{2} \)
71 \( 1 + 489.T + 3.57e5T^{2} \)
73 \( 1 - 511.T + 3.89e5T^{2} \)
79 \( 1 - 197.T + 4.93e5T^{2} \)
83 \( 1 - 391.T + 5.71e5T^{2} \)
89 \( 1 - 836.T + 7.04e5T^{2} \)
97 \( 1 + 1.19e3T + 9.12e5T^{2} \)
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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

−8.391740279022058842094010339990, −8.167512954505009176712441211556, −6.75729503397174483294930854444, −6.19427077845370418138135050901, −5.38442554797238713586899534382, −4.26471881663757917888497180590, −3.67601090501880165231940642592, −2.01720754889076613654874973210, −0.998814198239016407257136529250, 0, 0.998814198239016407257136529250, 2.01720754889076613654874973210, 3.67601090501880165231940642592, 4.26471881663757917888497180590, 5.38442554797238713586899534382, 6.19427077845370418138135050901, 6.75729503397174483294930854444, 8.167512954505009176712441211556, 8.391740279022058842094010339990

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