Properties

Degree $2$
Conductor $1849$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 5.33·2-s + 5.38·3-s + 20.4·4-s + 1.13·5-s − 28.6·6-s + 19.4·7-s − 66.3·8-s + 1.94·9-s − 6.07·10-s + 11.2·11-s + 109.·12-s + 82.3·13-s − 103.·14-s + 6.13·15-s + 190.·16-s − 132.·17-s − 10.3·18-s + 41.5·19-s + 23.2·20-s + 104.·21-s − 59.9·22-s + 97.0·23-s − 356.·24-s − 123.·25-s − 439.·26-s − 134.·27-s + 397.·28-s + ⋯
L(s)  = 1  − 1.88·2-s + 1.03·3-s + 2.55·4-s + 0.101·5-s − 1.95·6-s + 1.04·7-s − 2.93·8-s + 0.0722·9-s − 0.192·10-s + 0.308·11-s + 2.64·12-s + 1.75·13-s − 1.97·14-s + 0.105·15-s + 2.97·16-s − 1.89·17-s − 0.136·18-s + 0.501·19-s + 0.260·20-s + 1.08·21-s − 0.580·22-s + 0.879·23-s − 3.03·24-s − 0.989·25-s − 3.31·26-s − 0.960·27-s + 2.68·28-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 \( 1 + 5.33T + 8T^{2} \)
3 \( 1 - 5.38T + 27T^{2} \)
5 \( 1 - 1.13T + 125T^{2} \)
7 \( 1 - 19.4T + 343T^{2} \)
11 \( 1 - 11.2T + 1.33e3T^{2} \)
13 \( 1 - 82.3T + 2.19e3T^{2} \)
17 \( 1 + 132.T + 4.91e3T^{2} \)
19 \( 1 - 41.5T + 6.85e3T^{2} \)
23 \( 1 - 97.0T + 1.21e4T^{2} \)
29 \( 1 + 94.9T + 2.43e4T^{2} \)
31 \( 1 + 265.T + 2.97e4T^{2} \)
37 \( 1 + 70.0T + 5.06e4T^{2} \)
41 \( 1 - 30.4T + 6.89e4T^{2} \)
47 \( 1 + 418.T + 1.03e5T^{2} \)
53 \( 1 + 399.T + 1.48e5T^{2} \)
59 \( 1 + 57.1T + 2.05e5T^{2} \)
61 \( 1 - 331.T + 2.26e5T^{2} \)
67 \( 1 - 156.T + 3.00e5T^{2} \)
71 \( 1 + 564.T + 3.57e5T^{2} \)
73 \( 1 - 69.0T + 3.89e5T^{2} \)
79 \( 1 + 569.T + 4.93e5T^{2} \)
83 \( 1 + 785.T + 5.71e5T^{2} \)
89 \( 1 + 429.T + 7.04e5T^{2} \)
97 \( 1 + 1.16e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.538866849360190745640160075084, −8.135984371326979805906025301433, −7.30589682721544302603785981595, −6.52995235699399439499815602842, −5.57045617539605201830707674111, −4.00092639076621763893940110073, −2.99916465472256963067101413926, −1.87260936355694664845445596656, −1.49251940949962924416390441208, 0, 1.49251940949962924416390441208, 1.87260936355694664845445596656, 2.99916465472256963067101413926, 4.00092639076621763893940110073, 5.57045617539605201830707674111, 6.52995235699399439499815602842, 7.30589682721544302603785981595, 8.135984371326979805906025301433, 8.538866849360190745640160075084

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