↑

Properties

Label 2-68400-1.1-c1-0-0
Degree $2$
Conductor $68400$
Sign $1$
Analytic cond. $546.176$
Root an. cond. $23.3704$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 4·11-s − 4·17-s − 19-s − 6·23-s + 29-s − 4·31-s − 8·37-s − 5·41-s − 4·43-s + 6·47-s − 6·49-s − 3·53-s − 11·59-s − 13·61-s + 14·67-s + 15·71-s − 13·73-s − 4·77-s + 2·79-s + 14·83-s + 15·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.20·11-s − 0.970·17-s − 0.229·19-s − 1.25·23-s + 0.185·29-s − 0.718·31-s − 1.31·37-s − 0.780·41-s − 0.609·43-s + 0.875·47-s − 6/7·49-s − 0.412·53-s − 1.43·59-s − 1.66·61-s + 1.71·67-s + 1.78·71-s − 1.52·73-s − 0.455·77-s + 0.225·79-s + 1.53·83-s + 1.58·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(546.176\)
Root analytic conductor: \(23.3704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 68400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4255580375\)
\(L(\frac12)\) \(\approx\) \(0.4255580375\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−14.04277153545832, −13.68697817223111, −13.23251418343015, −12.66104416621023, −12.14886812432095, −11.75306214005650, −10.88826144229398, −10.77761518821575, −10.23813961271812, −9.590615721265915, −9.035226901703337, −8.501599990845126, −7.848745951232008, −7.731648392685038, −6.795633794352236, −6.432895749368516, −5.735690995079921, −5.038916530303295, −4.828280530838075, −3.962442107498226, −3.444910940056730, −2.619774102550217, −2.059575096635694, −1.490052329139196, −0.2050793860811671, 0.2050793860811671, 1.490052329139196, 2.059575096635694, 2.619774102550217, 3.444910940056730, 3.962442107498226, 4.828280530838075, 5.038916530303295, 5.735690995079921, 6.432895749368516, 6.795633794352236, 7.731648392685038, 7.848745951232008, 8.501599990845126, 9.035226901703337, 9.590615721265915, 10.23813961271812, 10.77761518821575, 10.88826144229398, 11.75306214005650, 12.14886812432095, 12.66104416621023, 13.23251418343015, 13.68697817223111, 14.04277153545832

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