Properties

Label 2-43e2-1.1-c3-0-271
Degree $2$
Conductor $1849$
Sign $-1$
Analytic cond. $109.094$
Root an. cond. $10.4448$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.94·2-s − 1.92·3-s − 4.21·4-s + 20.3·5-s + 3.74·6-s − 13.1·7-s + 23.7·8-s − 23.2·9-s − 39.6·10-s − 10.4·11-s + 8.10·12-s − 22.0·13-s + 25.4·14-s − 39.1·15-s − 12.5·16-s − 32.5·17-s + 45.3·18-s + 81.0·19-s − 85.8·20-s + 25.2·21-s + 20.4·22-s − 43.2·23-s − 45.7·24-s + 289.·25-s + 42.8·26-s + 96.7·27-s + 55.2·28-s + ⋯
L(s)  = 1  − 0.687·2-s − 0.370·3-s − 0.526·4-s + 1.82·5-s + 0.254·6-s − 0.707·7-s + 1.05·8-s − 0.862·9-s − 1.25·10-s − 0.287·11-s + 0.195·12-s − 0.470·13-s + 0.486·14-s − 0.674·15-s − 0.195·16-s − 0.463·17-s + 0.593·18-s + 0.978·19-s − 0.959·20-s + 0.261·21-s + 0.197·22-s − 0.391·23-s − 0.388·24-s + 2.31·25-s + 0.323·26-s + 0.689·27-s + 0.372·28-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: $-1$
Analytic conductor: \(109.094\)
Root analytic conductor: \(10.4448\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
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 \( 1 + 1.94T + 8T^{2} \)
3 \( 1 + 1.92T + 27T^{2} \)
5 \( 1 - 20.3T + 125T^{2} \)
7 \( 1 + 13.1T + 343T^{2} \)
11 \( 1 + 10.4T + 1.33e3T^{2} \)
13 \( 1 + 22.0T + 2.19e3T^{2} \)
17 \( 1 + 32.5T + 4.91e3T^{2} \)
19 \( 1 - 81.0T + 6.85e3T^{2} \)
23 \( 1 + 43.2T + 1.21e4T^{2} \)
29 \( 1 - 263.T + 2.43e4T^{2} \)
31 \( 1 + 269.T + 2.97e4T^{2} \)
37 \( 1 - 228.T + 5.06e4T^{2} \)
41 \( 1 - 23.1T + 6.89e4T^{2} \)
47 \( 1 - 143.T + 1.03e5T^{2} \)
53 \( 1 + 438.T + 1.48e5T^{2} \)
59 \( 1 + 446.T + 2.05e5T^{2} \)
61 \( 1 - 680.T + 2.26e5T^{2} \)
67 \( 1 - 91.4T + 3.00e5T^{2} \)
71 \( 1 + 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + 342.T + 3.89e5T^{2} \)
79 \( 1 - 816.T + 4.93e5T^{2} \)
83 \( 1 - 1.11e3T + 5.71e5T^{2} \)
89 \( 1 + 1.26e3T + 7.04e5T^{2} \)
97 \( 1 + 632.T + 9.12e5T^{2} \)
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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

−8.825953804510937473004741851537, −7.84937346140893723954684180494, −6.79775708663133134126620998195, −6.03730862967466596237057116308, −5.38655891834541798683400112652, −4.68037105832242891019631735409, −3.13898376053568923739654259722, −2.25062727960313333665546363277, −1.10324729745582311485164354926, 0, 1.10324729745582311485164354926, 2.25062727960313333665546363277, 3.13898376053568923739654259722, 4.68037105832242891019631735409, 5.38655891834541798683400112652, 6.03730862967466596237057116308, 6.79775708663133134126620998195, 7.84937346140893723954684180494, 8.825953804510937473004741851537

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