Properties

Label 1-419-419.418-r1-0-0
Degree $1$
Conductor $419$
Sign $1$
Analytic cond. $45.0278$
Root an. cond. $45.0278$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 419 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(419\)
Sign: $1$
Analytic conductor: \(45.0278\)
Root analytic conductor: \(45.0278\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{419} (418, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 419,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.726129511\)
\(L(\frac12)\) \(\approx\) \(2.726129511\)
\(L(1)\) \(\approx\) \(1.381291599\)
\(L(1)\) \(\approx\) \(1.381291599\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad419 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.284005557180463673177750092236, −23.59991986941719654124635127150, −21.74173284027447897314993654040, −20.964458544698368508897783318021, −20.76005016737446952923392486423, −19.66729936561195079783412960121, −18.595814318703163578469084574285, −18.07667759961316780571663801138, −17.33835620814682973152425457277, −16.13403524521494957231592263686, −15.266960337777193408520644949540, −14.48015986379768539146042678004, −13.43666788930377851317881739633, −12.6633797826727061403380517540, −10.876806805115837511376247367331, −10.67961657540470984865912612726, −9.32713513139667989058197898434, −8.72350533364957274307745017998, −7.9609634965533502906033284472, −6.923666381524386323244413387831, −5.81307400719492880729080547726, −4.44283222007722139186533808840, −2.78147512537896162074161363340, −2.09923184485817927515323486002, −1.09856054097616453791438539345, 1.09856054097616453791438539345, 2.09923184485817927515323486002, 2.78147512537896162074161363340, 4.44283222007722139186533808840, 5.81307400719492880729080547726, 6.923666381524386323244413387831, 7.9609634965533502906033284472, 8.72350533364957274307745017998, 9.32713513139667989058197898434, 10.67961657540470984865912612726, 10.876806805115837511376247367331, 12.6633797826727061403380517540, 13.43666788930377851317881739633, 14.48015986379768539146042678004, 15.266960337777193408520644949540, 16.13403524521494957231592263686, 17.33835620814682973152425457277, 18.07667759961316780571663801138, 18.595814318703163578469084574285, 19.66729936561195079783412960121, 20.76005016737446952923392486423, 20.964458544698368508897783318021, 21.74173284027447897314993654040, 23.59991986941719654124635127150, 24.284005557180463673177750092236

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