Properties

Label 2-1859-1.1-c1-0-114
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  + 1.50·2-s − 1.34·3-s + 0.267·4-s + 2.54·5-s − 2.01·6-s + 0.340·7-s − 2.60·8-s − 1.20·9-s + 3.83·10-s + 11-s − 0.359·12-s + 0.512·14-s − 3.40·15-s − 4.46·16-s − 6.84·17-s − 1.81·18-s − 2.05·19-s + 0.681·20-s − 0.456·21-s + 1.50·22-s − 6.76·23-s + 3.49·24-s + 1.46·25-s + 5.63·27-s + 0.0912·28-s − 1.25·29-s − 5.13·30-s + ⋯
L(s)  = 1  + 1.06·2-s − 0.774·3-s + 0.133·4-s + 1.13·5-s − 0.824·6-s + 0.128·7-s − 0.922·8-s − 0.400·9-s + 1.21·10-s + 0.301·11-s − 0.103·12-s + 0.137·14-s − 0.880·15-s − 1.11·16-s − 1.65·17-s − 0.426·18-s − 0.471·19-s + 0.152·20-s − 0.0996·21-s + 0.321·22-s − 1.41·23-s + 0.713·24-s + 0.293·25-s + 1.08·27-s + 0.0172·28-s − 0.233·29-s − 0.937·30-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 \( 1 - 1.50T + 2T^{2} \)
3 \( 1 + 1.34T + 3T^{2} \)
5 \( 1 - 2.54T + 5T^{2} \)
7 \( 1 - 0.340T + 7T^{2} \)
17 \( 1 + 6.84T + 17T^{2} \)
19 \( 1 + 2.05T + 19T^{2} \)
23 \( 1 + 6.76T + 23T^{2} \)
29 \( 1 + 1.25T + 29T^{2} \)
31 \( 1 + 1.07T + 31T^{2} \)
37 \( 1 - 8.12T + 37T^{2} \)
41 \( 1 - 7.18T + 41T^{2} \)
43 \( 1 + 4.43T + 43T^{2} \)
47 \( 1 + 0.783T + 47T^{2} \)
53 \( 1 + 8.45T + 53T^{2} \)
59 \( 1 - 8.65T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 + 5.40T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 6.66T + 73T^{2} \)
79 \( 1 + 0.0621T + 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 + 8.25T + 89T^{2} \)
97 \( 1 + 7.07T + 97T^{2} \)
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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

−9.032627506801127358345498599106, −8.108245875965740662467536324664, −6.68225712942762477633790424315, −6.09622378027650206888677692146, −5.76385545095574293489291491842, −4.74304511507371211544036196817, −4.19347569221223390779311463668, −2.85405860783575210837220800789, −1.91592893633203625821377348764, 0, 1.91592893633203625821377348764, 2.85405860783575210837220800789, 4.19347569221223390779311463668, 4.74304511507371211544036196817, 5.76385545095574293489291491842, 6.09622378027650206888677692146, 6.68225712942762477633790424315, 8.108245875965740662467536324664, 9.032627506801127358345498599106

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