↑

Properties

Label 2-1680-60.59-c1-0-9
Degree $2$
Conductor $1680$
Sign $-0.475 - 0.879i$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
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  + (1.54 + 0.788i)3-s + (−1.84 − 1.26i)5-s − 7-s + (1.75 + 2.43i)9-s − 0.392·11-s − 1.26i·13-s + (−1.84 − 3.40i)15-s − 5.68·17-s + 7.69i·19-s + (−1.54 − 0.788i)21-s − 0.620i·23-s + (1.78 + 4.66i)25-s + (0.786 + 5.13i)27-s + 6.78i·29-s + 4.17i·31-s + ⋯
L(s)  = 1  + (0.890 + 0.455i)3-s + (−0.823 − 0.566i)5-s − 0.377·7-s + (0.585 + 0.811i)9-s − 0.118·11-s − 0.350i·13-s + (−0.475 − 0.879i)15-s − 1.37·17-s + 1.76i·19-s + (−0.336 − 0.172i)21-s − 0.129i·23-s + (0.357 + 0.933i)25-s + (0.151 + 0.988i)27-s + 1.25i·29-s + 0.749i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.475 - 0.879i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -0.475 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.181044358\)
\(L(\frac12)\) \(\approx\) \(1.181044358\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.54 - 0.788i)T \)
5 \( 1 + (1.84 + 1.26i)T \)
7 \( 1 + T \)
good11 \( 1 + 0.392T + 11T^{2} \)
13 \( 1 + 1.26iT - 13T^{2} \)
17 \( 1 + 5.68T + 17T^{2} \)
19 \( 1 - 7.69iT - 19T^{2} \)
23 \( 1 + 0.620iT - 23T^{2} \)
29 \( 1 - 6.78iT - 29T^{2} \)
31 \( 1 - 4.17iT - 31T^{2} \)
37 \( 1 + 3.52iT - 37T^{2} \)
41 \( 1 - 5.15iT - 41T^{2} \)
43 \( 1 - 6.59T + 43T^{2} \)
47 \( 1 - 2.95iT - 47T^{2} \)
53 \( 1 + 6.07T + 53T^{2} \)
59 \( 1 + 3.99T + 59T^{2} \)
61 \( 1 + 4.16T + 61T^{2} \)
67 \( 1 + 7.02T + 67T^{2} \)
71 \( 1 - 9.75T + 71T^{2} \)
73 \( 1 - 16.0iT - 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + 14.8iT - 83T^{2} \)
89 \( 1 - 5.28iT - 89T^{2} \)
97 \( 1 - 4.54iT - 97T^{2} \)
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

−9.342462074572811437876416322203, −8.876283247129372954326597123475, −8.077165037900584475973882210578, −7.53798655106284697815689264677, −6.51298510134370662152806502347, −5.32488261241491774745525652655, −4.44494451312105355787746782654, −3.73484260763734960851769023215, −2.88924242117997001476552257755, −1.54791001218893488392827193862, 0.38982214562598893331919113285, 2.23403142753295855848037709313, 2.89360054330096548754318651577, 3.96591234746542413661167878612, 4.62164857252652094208704339328, 6.24053720244238440288530309283, 6.84184832375695982129468564326, 7.47377125449768112399197137167, 8.264747699417951126753903092290, 9.059971458702368853807035695361

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