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  − 1.26·2-s − 9.40·3-s − 6.40·4-s − 0.325·5-s + 11.8·6-s − 8.80·7-s + 18.1·8-s + 61.3·9-s + 0.410·10-s − 23.6·11-s + 60.1·12-s − 23.2·13-s + 11.1·14-s + 3.05·15-s + 28.2·16-s − 125.·17-s − 77.5·18-s − 16.8·19-s + 2.08·20-s + 82.7·21-s + 29.9·22-s + 127.·23-s − 171.·24-s − 124.·25-s + 29.3·26-s − 322.·27-s + 56.3·28-s + ⋯
L(s)  = 1  − 0.446·2-s − 1.80·3-s − 0.800·4-s − 0.0290·5-s + 0.808·6-s − 0.475·7-s + 0.804·8-s + 2.27·9-s + 0.0129·10-s − 0.648·11-s + 1.44·12-s − 0.496·13-s + 0.212·14-s + 0.0526·15-s + 0.441·16-s − 1.79·17-s − 1.01·18-s − 0.203·19-s + 0.0232·20-s + 0.860·21-s + 0.289·22-s + 1.15·23-s − 1.45·24-s − 0.999·25-s + 0.221·26-s − 2.30·27-s + 0.380·28-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 \( 1 + 1.26T + 8T^{2} \)
3 \( 1 + 9.40T + 27T^{2} \)
5 \( 1 + 0.325T + 125T^{2} \)
7 \( 1 + 8.80T + 343T^{2} \)
11 \( 1 + 23.6T + 1.33e3T^{2} \)
13 \( 1 + 23.2T + 2.19e3T^{2} \)
17 \( 1 + 125.T + 4.91e3T^{2} \)
19 \( 1 + 16.8T + 6.85e3T^{2} \)
23 \( 1 - 127.T + 1.21e4T^{2} \)
29 \( 1 - 220.T + 2.43e4T^{2} \)
31 \( 1 + 303.T + 2.97e4T^{2} \)
37 \( 1 + 245.T + 5.06e4T^{2} \)
41 \( 1 - 185.T + 6.89e4T^{2} \)
47 \( 1 + 1.86T + 1.03e5T^{2} \)
53 \( 1 - 261.T + 1.48e5T^{2} \)
59 \( 1 + 417.T + 2.05e5T^{2} \)
61 \( 1 - 699.T + 2.26e5T^{2} \)
67 \( 1 + 279.T + 3.00e5T^{2} \)
71 \( 1 - 675.T + 3.57e5T^{2} \)
73 \( 1 - 1.02e3T + 3.89e5T^{2} \)
79 \( 1 + 732.T + 4.93e5T^{2} \)
83 \( 1 - 1.09e3T + 5.71e5T^{2} \)
89 \( 1 - 200.T + 7.04e5T^{2} \)
97 \( 1 - 1.44e3T + 9.12e5T^{2} \)
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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

−8.655974115401117536905915124538, −7.51073888753211119168387702939, −6.86011598439455820381766184301, −6.07723459137498591222561630147, −5.08946495603802623140250087850, −4.76120812625282337138215126697, −3.74548529317092745453985563978, −2.04335191575242413818340801297, −0.67073495555897230798352543142, 0, 0.67073495555897230798352543142, 2.04335191575242413818340801297, 3.74548529317092745453985563978, 4.76120812625282337138215126697, 5.08946495603802623140250087850, 6.07723459137498591222561630147, 6.86011598439455820381766184301, 7.51073888753211119168387702939, 8.655974115401117536905915124538

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