↑

Properties

Label 2-1920-80.29-c1-0-31
Degree $2$
Conductor $1920$
Sign $0.375 + 0.926i$
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 + (−2.16 − 0.561i)5-s + 4.51·7-s − 1.00i·9-s + (3.44 − 3.44i)11-s + (0.113 − 0.113i)13-s + (1.92 − 1.13i)15-s − 5.03i·17-s + (0.992 + 0.992i)19-s + (−3.19 + 3.19i)21-s − 8.00·23-s + (4.36 + 2.43i)25-s + (0.707 + 0.707i)27-s + (1.01 + 1.01i)29-s − 6.42·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.967 − 0.251i)5-s + 1.70·7-s − 0.333i·9-s + (1.03 − 1.03i)11-s + (0.0315 − 0.0315i)13-s + (0.497 − 0.292i)15-s − 1.22i·17-s + (0.227 + 0.227i)19-s + (−0.696 + 0.696i)21-s − 1.66·23-s + (0.873 + 0.486i)25-s + (0.136 + 0.136i)27-s + (0.188 + 0.188i)29-s − 1.15·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.375 + 0.926i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.375 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.352617906\)
\(L(\frac12)\) \(\approx\) \(1.352617906\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 + 0.561i)T \)
good7 \( 1 - 4.51T + 7T^{2} \)
11 \( 1 + (-3.44 + 3.44i)T - 11iT^{2} \)
13 \( 1 + (-0.113 + 0.113i)T - 13iT^{2} \)
17 \( 1 + 5.03iT - 17T^{2} \)
19 \( 1 + (-0.992 - 0.992i)T + 19iT^{2} \)
23 \( 1 + 8.00T + 23T^{2} \)
29 \( 1 + (-1.01 - 1.01i)T + 29iT^{2} \)
31 \( 1 + 6.42T + 31T^{2} \)
37 \( 1 + (1.63 + 1.63i)T + 37iT^{2} \)
41 \( 1 - 3.35iT - 41T^{2} \)
43 \( 1 + (5.68 + 5.68i)T + 43iT^{2} \)
47 \( 1 + 9.10iT - 47T^{2} \)
53 \( 1 + (3.27 + 3.27i)T + 53iT^{2} \)
59 \( 1 + (-5.30 + 5.30i)T - 59iT^{2} \)
61 \( 1 + (-5.87 - 5.87i)T + 61iT^{2} \)
67 \( 1 + (-1.87 + 1.87i)T - 67iT^{2} \)
71 \( 1 + 0.635iT - 71T^{2} \)
73 \( 1 - 6.14T + 73T^{2} \)
79 \( 1 - 1.76T + 79T^{2} \)
83 \( 1 + (-6.39 + 6.39i)T - 83iT^{2} \)
89 \( 1 - 0.579iT - 89T^{2} \)
97 \( 1 + 15.2iT - 97T^{2} \)
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

−8.816586889280649077101645518653, −8.370949276046268810798153935525, −7.60847674659402956599131168837, −6.76961544197178324973236078427, −5.57856912242713727396459985774, −5.00465650534786803584422993103, −4.10436452445601417946372046816, −3.48378650230300214345774966306, −1.80802722045827381459727389485, −0.57272665527330420346727624344, 1.33967165118803015220999286158, 2.10715408595516446027764981061, 3.83975333339097754785439187234, 4.35754466087445467881944137824, 5.22337239183101067812261003352, 6.30561550378442532323401596551, 7.07664468904538922955715887119, 7.943302250061933486758136075872, 8.165291054648832455669498935854, 9.247170138487860654869005984190

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