Properties

Label 2-1859-1.1-c1-0-121
Degree $2$
Conductor $1859$
Sign $-1$
Analytic cond. $14.8441$
Root an. cond. $3.85281$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.49·2-s + 0.384·3-s + 0.241·4-s + 2.16·5-s + 0.575·6-s − 2.79·7-s − 2.63·8-s − 2.85·9-s + 3.24·10-s − 11-s + 0.0927·12-s − 4.18·14-s + 0.832·15-s − 4.42·16-s − 3.44·17-s − 4.27·18-s + 0.00264·19-s + 0.523·20-s − 1.07·21-s − 1.49·22-s + 7.04·23-s − 1.01·24-s − 0.299·25-s − 2.24·27-s − 0.674·28-s − 3.93·29-s + 1.24·30-s + ⋯
L(s)  = 1  + 1.05·2-s + 0.221·3-s + 0.120·4-s + 0.969·5-s + 0.234·6-s − 1.05·7-s − 0.930·8-s − 0.950·9-s + 1.02·10-s − 0.301·11-s + 0.0267·12-s − 1.11·14-s + 0.215·15-s − 1.10·16-s − 0.834·17-s − 1.00·18-s + 0.000606·19-s + 0.117·20-s − 0.234·21-s − 0.319·22-s + 1.46·23-s − 0.206·24-s − 0.0599·25-s − 0.432·27-s − 0.127·28-s − 0.730·29-s + 0.227·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(14.8441\)
Root analytic conductor: \(3.85281\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 1.49T + 2T^{2} \)
3 \( 1 - 0.384T + 3T^{2} \)
5 \( 1 - 2.16T + 5T^{2} \)
7 \( 1 + 2.79T + 7T^{2} \)
17 \( 1 + 3.44T + 17T^{2} \)
19 \( 1 - 0.00264T + 19T^{2} \)
23 \( 1 - 7.04T + 23T^{2} \)
29 \( 1 + 3.93T + 29T^{2} \)
31 \( 1 + 1.20T + 31T^{2} \)
37 \( 1 - 1.77T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 8.73T + 53T^{2} \)
59 \( 1 - 5.43T + 59T^{2} \)
61 \( 1 + 4.14T + 61T^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 - 0.934T + 71T^{2} \)
73 \( 1 - 2.93T + 73T^{2} \)
79 \( 1 - 2.39T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 6.54T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−9.024809544934492137279484115742, −8.219211090669705210320907394833, −6.73111879234350852278166299249, −6.41190414380979492287899609064, −5.40796079514082816269453749600, −5.01268828787392738602891457427, −3.62557067003437643797314319299, −3.05727681247327257001718026056, −2.12839767088643104061709464304, 0, 2.12839767088643104061709464304, 3.05727681247327257001718026056, 3.62557067003437643797314319299, 5.01268828787392738602891457427, 5.40796079514082816269453749600, 6.41190414380979492287899609064, 6.73111879234350852278166299249, 8.219211090669705210320907394833, 9.024809544934492137279484115742

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