↑

Properties

Label 2-1920-80.69-c1-0-20
Degree $2$
Conductor $1920$
Sign $0.608 - 0.793i$
Analytic cond. $15.3312$
Root an. cond. $3.91551$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)3-s + (1.24 + 1.86i)5-s + 1.58·7-s + 1.00i·9-s + (3.92 + 3.92i)11-s + (3.10 + 3.10i)13-s + (0.438 − 2.19i)15-s − 1.48i·17-s + (4.94 − 4.94i)19-s + (−1.12 − 1.12i)21-s − 6.61·23-s + (−1.92 + 4.61i)25-s + (0.707 − 0.707i)27-s + (−4.42 + 4.42i)29-s + 1.50·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.554 + 0.831i)5-s + 0.600·7-s + 0.333i·9-s + (1.18 + 1.18i)11-s + (0.861 + 0.861i)13-s + (0.113 − 0.566i)15-s − 0.359i·17-s + (1.13 − 1.13i)19-s + (−0.245 − 0.245i)21-s − 1.38·23-s + (−0.384 + 0.923i)25-s + (0.136 − 0.136i)27-s + (−0.821 + 0.821i)29-s + 0.270·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.608 - 0.793i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.966981434\)
\(L(\frac12)\) \(\approx\) \(1.966981434\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.24 - 1.86i)T \)
good7 \( 1 - 1.58T + 7T^{2} \)
11 \( 1 + (-3.92 - 3.92i)T + 11iT^{2} \)
13 \( 1 + (-3.10 - 3.10i)T + 13iT^{2} \)
17 \( 1 + 1.48iT - 17T^{2} \)
19 \( 1 + (-4.94 + 4.94i)T - 19iT^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 + (4.42 - 4.42i)T - 29iT^{2} \)
31 \( 1 - 1.50T + 31T^{2} \)
37 \( 1 + (2.14 - 2.14i)T - 37iT^{2} \)
41 \( 1 - 6.84iT - 41T^{2} \)
43 \( 1 + (0.322 - 0.322i)T - 43iT^{2} \)
47 \( 1 + 13.3iT - 47T^{2} \)
53 \( 1 + (0.931 - 0.931i)T - 53iT^{2} \)
59 \( 1 + (-1.14 - 1.14i)T + 59iT^{2} \)
61 \( 1 + (2.67 - 2.67i)T - 61iT^{2} \)
67 \( 1 + (-5.43 - 5.43i)T + 67iT^{2} \)
71 \( 1 - 2.26iT - 71T^{2} \)
73 \( 1 - 5.27T + 73T^{2} \)
79 \( 1 - 6.52T + 79T^{2} \)
83 \( 1 + (-0.973 - 0.973i)T + 83iT^{2} \)
89 \( 1 + 6.83iT - 89T^{2} \)
97 \( 1 + 15.8iT - 97T^{2} \)
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

−9.460181076052226134995040966766, −8.583718922134235602442271961930, −7.44574496926275828573231774847, −6.90498699748598653661729098198, −6.35119373996158834281221680248, −5.36268398342776705728778596439, −4.47174635875582687048404520273, −3.45937041555633063734332250195, −2.09922397444429691645223781113, −1.42684307971873142563984580809, 0.842489583170047178177985484563, 1.75236890022460217009923921181, 3.49268396032295007851737976859, 4.05484809105389434333594945179, 5.22052160500357208510038558937, 5.91132790850658498070807229463, 6.23998693865189474759566928489, 7.88454261655721879176745676740, 8.253805257716116904249252687524, 9.188246533457682963024370477908

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