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  − 0.902·2-s − 9.58·3-s − 7.18·4-s − 2.03·5-s + 8.65·6-s + 26.8·7-s + 13.7·8-s + 64.8·9-s + 1.83·10-s + 38.6·11-s + 68.8·12-s − 37.4·13-s − 24.2·14-s + 19.4·15-s + 45.1·16-s − 12.0·17-s − 58.5·18-s − 64.0·19-s + 14.6·20-s − 257.·21-s − 34.9·22-s − 85.2·23-s − 131.·24-s − 120.·25-s + 33.8·26-s − 362.·27-s − 192.·28-s + ⋯
L(s)  = 1  − 0.319·2-s − 1.84·3-s − 0.898·4-s − 0.181·5-s + 0.588·6-s + 1.44·7-s + 0.605·8-s + 2.40·9-s + 0.0580·10-s + 1.06·11-s + 1.65·12-s − 0.799·13-s − 0.462·14-s + 0.335·15-s + 0.704·16-s − 0.172·17-s − 0.766·18-s − 0.773·19-s + 0.163·20-s − 2.67·21-s − 0.338·22-s − 0.773·23-s − 1.11·24-s − 0.966·25-s + 0.255·26-s − 2.58·27-s − 1.30·28-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 \( 1 + 0.902T + 8T^{2} \)
3 \( 1 + 9.58T + 27T^{2} \)
5 \( 1 + 2.03T + 125T^{2} \)
7 \( 1 - 26.8T + 343T^{2} \)
11 \( 1 - 38.6T + 1.33e3T^{2} \)
13 \( 1 + 37.4T + 2.19e3T^{2} \)
17 \( 1 + 12.0T + 4.91e3T^{2} \)
19 \( 1 + 64.0T + 6.85e3T^{2} \)
23 \( 1 + 85.2T + 1.21e4T^{2} \)
29 \( 1 - 129.T + 2.43e4T^{2} \)
31 \( 1 - 261.T + 2.97e4T^{2} \)
37 \( 1 - 310.T + 5.06e4T^{2} \)
41 \( 1 + 333.T + 6.89e4T^{2} \)
47 \( 1 + 475.T + 1.03e5T^{2} \)
53 \( 1 - 14.1T + 1.48e5T^{2} \)
59 \( 1 + 652.T + 2.05e5T^{2} \)
61 \( 1 + 233.T + 2.26e5T^{2} \)
67 \( 1 - 963.T + 3.00e5T^{2} \)
71 \( 1 - 887.T + 3.57e5T^{2} \)
73 \( 1 + 345.T + 3.89e5T^{2} \)
79 \( 1 + 396.T + 4.93e5T^{2} \)
83 \( 1 - 539.T + 5.71e5T^{2} \)
89 \( 1 - 295.T + 7.04e5T^{2} \)
97 \( 1 - 191.T + 9.12e5T^{2} \)
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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

−8.293744457501965285942258014914, −7.85421422104862583444943772344, −6.73541464155212468829890634690, −6.06780256284627150397582310962, −5.02849149094558640105812586546, −4.61460436890054248943260895693, −4.03775548988269611213185283699, −1.81584072821086226435019661451, −0.976167151364885328108151243550, 0, 0.976167151364885328108151243550, 1.81584072821086226435019661451, 4.03775548988269611213185283699, 4.61460436890054248943260895693, 5.02849149094558640105812586546, 6.06780256284627150397582310962, 6.73541464155212468829890634690, 7.85421422104862583444943772344, 8.293744457501965285942258014914

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