Properties

Label 2-1859-1.1-c1-0-6
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.22·2-s − 2.50·3-s − 0.507·4-s − 2.28·5-s + 3.06·6-s − 0.285·7-s + 3.06·8-s + 3.28·9-s + 2.79·10-s + 11-s + 1.27·12-s + 0.348·14-s + 5.72·15-s − 2.72·16-s + 0.507·17-s − 4.01·18-s − 3.95·19-s + 1.15·20-s + 0.714·21-s − 1.22·22-s − 0.0632·23-s − 7.67·24-s + 0.221·25-s − 0.714·27-s + 0.144·28-s − 10.7·29-s − 6.99·30-s + ⋯
L(s)  = 1  − 0.863·2-s − 1.44·3-s − 0.253·4-s − 1.02·5-s + 1.25·6-s − 0.107·7-s + 1.08·8-s + 1.09·9-s + 0.882·10-s + 0.301·11-s + 0.366·12-s + 0.0931·14-s + 1.47·15-s − 0.682·16-s + 0.122·17-s − 0.946·18-s − 0.906·19-s + 0.259·20-s + 0.155·21-s − 0.260·22-s − 0.0131·23-s − 1.56·24-s + 0.0443·25-s − 0.137·27-s + 0.0273·28-s − 1.99·29-s − 1.27·30-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: \(0\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1576520492\)
\(L(\frac12)\) \(\approx\) \(0.1576520492\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.22T + 2T^{2} \)
3 \( 1 + 2.50T + 3T^{2} \)
5 \( 1 + 2.28T + 5T^{2} \)
7 \( 1 + 0.285T + 7T^{2} \)
17 \( 1 - 0.507T + 17T^{2} \)
19 \( 1 + 3.95T + 19T^{2} \)
23 \( 1 + 0.0632T + 23T^{2} \)
29 \( 1 + 10.7T + 29T^{2} \)
31 \( 1 + 3.57T + 31T^{2} \)
37 \( 1 - 7.72T + 37T^{2} \)
41 \( 1 + 0.380T + 41T^{2} \)
43 \( 1 + 5.41T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + 6.72T + 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 + 3.33T + 61T^{2} \)
67 \( 1 - 0.207T + 67T^{2} \)
71 \( 1 + 2.77T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 3.45T + 79T^{2} \)
83 \( 1 - 4.74T + 83T^{2} \)
89 \( 1 + 3.71T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.344679373371100768254945024322, −8.385990271564023587612110557169, −7.73863984422679575069635104497, −6.96716370258564883128638226459, −6.12574649355811407944243456199, −5.17346334210732489632022166337, −4.39906325846531694650058762205, −3.62490759439113871721964423050, −1.66899536452310608264532338416, −0.33540848726604619649658509000, 0.33540848726604619649658509000, 1.66899536452310608264532338416, 3.62490759439113871721964423050, 4.39906325846531694650058762205, 5.17346334210732489632022166337, 6.12574649355811407944243456199, 6.96716370258564883128638226459, 7.73863984422679575069635104497, 8.385990271564023587612110557169, 9.344679373371100768254945024322

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