↑

Properties

Label 2-1920-1920.869-c0-0-0
Degree $2$
Conductor $1920$
Sign $-0.336 - 0.941i$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0980 + 0.995i)2-s + (−0.881 + 0.471i)3-s + (−0.980 + 0.195i)4-s + (−0.634 − 0.773i)5-s + (−0.555 − 0.831i)6-s + (−0.290 − 0.956i)8-s + (0.555 − 0.831i)9-s + (0.707 − 0.707i)10-s + (0.773 − 0.634i)12-s + (0.923 + 0.382i)15-s + (0.923 − 0.382i)16-s + (−1.42 + 0.591i)17-s + (0.881 + 0.471i)18-s + (0.0569 + 0.577i)19-s + (0.773 + 0.634i)20-s + ⋯
L(s)  = 1  + (0.0980 + 0.995i)2-s + (−0.881 + 0.471i)3-s + (−0.980 + 0.195i)4-s + (−0.634 − 0.773i)5-s + (−0.555 − 0.831i)6-s + (−0.290 − 0.956i)8-s + (0.555 − 0.831i)9-s + (0.707 − 0.707i)10-s + (0.773 − 0.634i)12-s + (0.923 + 0.382i)15-s + (0.923 − 0.382i)16-s + (−1.42 + 0.591i)17-s + (0.881 + 0.471i)18-s + (0.0569 + 0.577i)19-s + (0.773 + 0.634i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.336 - 0.941i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :0),\ -0.336 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6333812820\)
\(L(\frac12)\) \(\approx\) \(0.6333812820\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0980 - 0.995i)T \)
3 \( 1 + (0.881 - 0.471i)T \)
5 \( 1 + (0.634 + 0.773i)T \)
good7 \( 1 + (-0.382 + 0.923i)T^{2} \)
11 \( 1 + (-0.831 + 0.555i)T^{2} \)
13 \( 1 + (-0.195 - 0.980i)T^{2} \)
17 \( 1 + (1.42 - 0.591i)T + (0.707 - 0.707i)T^{2} \)
19 \( 1 + (-0.0569 - 0.577i)T + (-0.980 + 0.195i)T^{2} \)
23 \( 1 + (-1.95 + 0.388i)T + (0.923 - 0.382i)T^{2} \)
29 \( 1 + (0.831 + 0.555i)T^{2} \)
31 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
37 \( 1 + (-0.980 - 0.195i)T^{2} \)
41 \( 1 + (0.923 - 0.382i)T^{2} \)
43 \( 1 + (-0.555 - 0.831i)T^{2} \)
47 \( 1 + (-0.360 - 0.871i)T + (-0.707 + 0.707i)T^{2} \)
53 \( 1 + (-0.482 - 1.59i)T + (-0.831 + 0.555i)T^{2} \)
59 \( 1 + (-0.195 + 0.980i)T^{2} \)
61 \( 1 + (0.902 + 1.68i)T + (-0.555 + 0.831i)T^{2} \)
67 \( 1 + (0.555 - 0.831i)T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (-0.382 - 0.923i)T^{2} \)
79 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (-1.10 + 0.108i)T + (0.980 - 0.195i)T^{2} \)
89 \( 1 + (-0.923 - 0.382i)T^{2} \)
97 \( 1 - iT^{2} \)
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

−9.298104412442480327406953190608, −8.846693067792340457513167805257, −8.098320500075411699156086618145, −7.04795167447869546791814839851, −6.51399269518485686190996303321, −5.57127102701518236370363054118, −4.74413329959974018318998005211, −4.37481987481558273003645893690, −3.32136786383683610576539564158, −1.02593860001278301409310209096, 0.67192998363772249442507512794, 2.23392221005704814104657497936, 3.03142237527126025229764219075, 4.30818301354173097270847307414, 4.83812149343785273933282957036, 5.92476136027861602146683140834, 6.84191546010076660809011517433, 7.43760935020406202507174135083, 8.477717408379870589509414884576, 9.293595126987423453577583447764

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