Properties

Label 2-1859-1.1-c1-0-12
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.340·2-s − 1.21·3-s − 1.88·4-s − 3.18·5-s + 0.413·6-s + 4.05·7-s + 1.32·8-s − 1.52·9-s + 1.08·10-s + 11-s + 2.28·12-s − 1.38·14-s + 3.86·15-s + 3.31·16-s − 2.34·17-s + 0.520·18-s − 6.23·19-s + 6.00·20-s − 4.91·21-s − 0.340·22-s − 8.20·23-s − 1.60·24-s + 5.16·25-s + 5.49·27-s − 7.63·28-s − 4.01·29-s − 1.31·30-s + ⋯
L(s)  = 1  − 0.240·2-s − 0.700·3-s − 0.942·4-s − 1.42·5-s + 0.168·6-s + 1.53·7-s + 0.467·8-s − 0.509·9-s + 0.343·10-s + 0.301·11-s + 0.659·12-s − 0.368·14-s + 0.998·15-s + 0.829·16-s − 0.569·17-s + 0.122·18-s − 1.43·19-s + 1.34·20-s − 1.07·21-s − 0.0725·22-s − 1.71·23-s − 0.327·24-s + 1.03·25-s + 1.05·27-s − 1.44·28-s − 0.745·29-s − 0.240·30-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

\(L(1)\) \(\approx\) \(0.4410187643\)
\(L(\frac12)\) \(\approx\) \(0.4410187643\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.340T + 2T^{2} \)
3 \( 1 + 1.21T + 3T^{2} \)
5 \( 1 + 3.18T + 5T^{2} \)
7 \( 1 - 4.05T + 7T^{2} \)
17 \( 1 + 2.34T + 17T^{2} \)
19 \( 1 + 6.23T + 19T^{2} \)
23 \( 1 + 8.20T + 23T^{2} \)
29 \( 1 + 4.01T + 29T^{2} \)
31 \( 1 - 0.723T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 + 4.40T + 41T^{2} \)
43 \( 1 - 9.58T + 43T^{2} \)
47 \( 1 - 4.76T + 47T^{2} \)
53 \( 1 - 5.18T + 53T^{2} \)
59 \( 1 + 0.694T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 7.17T + 67T^{2} \)
71 \( 1 - 2.32T + 71T^{2} \)
73 \( 1 + 2.44T + 73T^{2} \)
79 \( 1 + 2.59T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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

−8.832530412643319931020570507818, −8.451022174587057553719704327696, −7.935190000774887074981223668209, −7.05616261066736556802834523529, −5.85839539700578905969881343322, −5.04512864984926395439599602655, −4.26442876486035095977204877974, −3.83664933194507851143434318304, −1.99311065859564579050387061835, −0.47936065085272333358985199819, 0.47936065085272333358985199819, 1.99311065859564579050387061835, 3.83664933194507851143434318304, 4.26442876486035095977204877974, 5.04512864984926395439599602655, 5.85839539700578905969881343322, 7.05616261066736556802834523529, 7.935190000774887074981223668209, 8.451022174587057553719704327696, 8.832530412643319931020570507818

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