Properties

Label 2-1859-1.1-c1-0-49
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.13·2-s − 2.20·3-s + 2.56·4-s − 1.38·5-s − 4.71·6-s + 5.12·7-s + 1.19·8-s + 1.88·9-s − 2.95·10-s + 11-s − 5.65·12-s + 10.9·14-s + 3.05·15-s − 2.56·16-s − 0.645·17-s + 4.02·18-s + 5.71·19-s − 3.54·20-s − 11.3·21-s + 2.13·22-s − 2.40·23-s − 2.64·24-s − 3.08·25-s + 2.46·27-s + 13.1·28-s + 2.88·29-s + 6.53·30-s + ⋯
L(s)  = 1  + 1.51·2-s − 1.27·3-s + 1.28·4-s − 0.619·5-s − 1.92·6-s + 1.93·7-s + 0.423·8-s + 0.627·9-s − 0.935·10-s + 0.301·11-s − 1.63·12-s + 2.92·14-s + 0.789·15-s − 0.640·16-s − 0.156·17-s + 0.947·18-s + 1.31·19-s − 0.792·20-s − 2.46·21-s + 0.455·22-s − 0.502·23-s − 0.540·24-s − 0.616·25-s + 0.475·27-s + 2.47·28-s + 0.535·29-s + 1.19·30-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: $\chi_{1859} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.858417238\)
\(L(\frac12)\) \(\approx\) \(2.858417238\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.13T + 2T^{2} \)
3 \( 1 + 2.20T + 3T^{2} \)
5 \( 1 + 1.38T + 5T^{2} \)
7 \( 1 - 5.12T + 7T^{2} \)
17 \( 1 + 0.645T + 17T^{2} \)
19 \( 1 - 5.71T + 19T^{2} \)
23 \( 1 + 2.40T + 23T^{2} \)
29 \( 1 - 2.88T + 29T^{2} \)
31 \( 1 - 3.43T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + 3.99T + 41T^{2} \)
43 \( 1 - 7.08T + 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + 0.0415T + 59T^{2} \)
61 \( 1 + 4.29T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 4.10T + 71T^{2} \)
73 \( 1 - 4.85T + 73T^{2} \)
79 \( 1 - 3.30T + 79T^{2} \)
83 \( 1 - 3.87T + 83T^{2} \)
89 \( 1 - 5.70T + 89T^{2} \)
97 \( 1 + 0.622T + 97T^{2} \)
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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

−9.257655568233209303134402890571, −8.077696835388999741381660392469, −7.53556280079850727748970252350, −6.46959791428397762718342950617, −5.73480308809142319978637079418, −5.06379690482168320467928826013, −4.54806076760642776096793415344, −3.83287943339158845369428986348, −2.43853530358166760607933408190, −1.03601625879472138094505889658, 1.03601625879472138094505889658, 2.43853530358166760607933408190, 3.83287943339158845369428986348, 4.54806076760642776096793415344, 5.06379690482168320467928826013, 5.73480308809142319978637079418, 6.46959791428397762718342950617, 7.53556280079850727748970252350, 8.077696835388999741381660392469, 9.257655568233209303134402890571

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