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  + 4.78·2-s − 1.15·3-s + 14.8·4-s + 21.3·5-s − 5.53·6-s − 24.0·7-s + 32.7·8-s − 25.6·9-s + 101.·10-s + 17.5·11-s − 17.1·12-s − 56.9·13-s − 114.·14-s − 24.6·15-s + 37.8·16-s − 74.3·17-s − 122.·18-s − 100.·19-s + 316.·20-s + 27.8·21-s + 84.1·22-s − 13.9·23-s − 37.9·24-s + 328.·25-s − 272.·26-s + 60.9·27-s − 357.·28-s + ⋯
L(s)  = 1  + 1.69·2-s − 0.222·3-s + 1.85·4-s + 1.90·5-s − 0.376·6-s − 1.29·7-s + 1.44·8-s − 0.950·9-s + 3.22·10-s + 0.482·11-s − 0.413·12-s − 1.21·13-s − 2.19·14-s − 0.424·15-s + 0.591·16-s − 1.06·17-s − 1.60·18-s − 1.21·19-s + 3.53·20-s + 0.289·21-s + 0.815·22-s − 0.126·23-s − 0.322·24-s + 2.63·25-s − 2.05·26-s + 0.434·27-s − 2.41·28-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 \( 1 - 4.78T + 8T^{2} \)
3 \( 1 + 1.15T + 27T^{2} \)
5 \( 1 - 21.3T + 125T^{2} \)
7 \( 1 + 24.0T + 343T^{2} \)
11 \( 1 - 17.5T + 1.33e3T^{2} \)
13 \( 1 + 56.9T + 2.19e3T^{2} \)
17 \( 1 + 74.3T + 4.91e3T^{2} \)
19 \( 1 + 100.T + 6.85e3T^{2} \)
23 \( 1 + 13.9T + 1.21e4T^{2} \)
29 \( 1 + 87.1T + 2.43e4T^{2} \)
31 \( 1 + 127.T + 2.97e4T^{2} \)
37 \( 1 + 125.T + 5.06e4T^{2} \)
41 \( 1 + 82.7T + 6.89e4T^{2} \)
47 \( 1 + 71.5T + 1.03e5T^{2} \)
53 \( 1 - 152.T + 1.48e5T^{2} \)
59 \( 1 - 319.T + 2.05e5T^{2} \)
61 \( 1 - 353.T + 2.26e5T^{2} \)
67 \( 1 - 37.2T + 3.00e5T^{2} \)
71 \( 1 - 781.T + 3.57e5T^{2} \)
73 \( 1 + 1.17e3T + 3.89e5T^{2} \)
79 \( 1 - 841.T + 4.93e5T^{2} \)
83 \( 1 + 1.11e3T + 5.71e5T^{2} \)
89 \( 1 + 308.T + 7.04e5T^{2} \)
97 \( 1 - 1.10e3T + 9.12e5T^{2} \)
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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

−8.766719474470250813709058509457, −6.89119591867349598874489572926, −6.62978742149947101889743116097, −5.88837204771619008054739251580, −5.41642405707258241150737584502, −4.54045210985518340913860366622, −3.40068374200649222971859318730, −2.48835557367161954048494007245, −2.04756605907373287221339710053, 0, 2.04756605907373287221339710053, 2.48835557367161954048494007245, 3.40068374200649222971859318730, 4.54045210985518340913860366622, 5.41642405707258241150737584502, 5.88837204771619008054739251580, 6.62978742149947101889743116097, 6.89119591867349598874489572926, 8.766719474470250813709058509457

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