↑

Properties

Label 2-840-35.13-c1-0-16
Degree $2$
Conductor $840$
Sign $-0.412 + 0.911i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
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 + (−2.21 − 0.315i)5-s + (2.54 − 0.706i)7-s − 1.00i·9-s − 1.16·11-s + (−1.47 + 1.47i)13-s + (1.78 − 1.34i)15-s + (−1.17 − 1.17i)17-s − 2.09·19-s + (−1.30 + 2.30i)21-s + (−5.33 − 5.33i)23-s + (4.80 + 1.39i)25-s + (0.707 + 0.707i)27-s − 1.55i·29-s − 2.66i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.990 − 0.141i)5-s + (0.963 − 0.266i)7-s − 0.333i·9-s − 0.352·11-s + (−0.409 + 0.409i)13-s + (0.461 − 0.346i)15-s + (−0.285 − 0.285i)17-s − 0.481·19-s + (−0.284 + 0.502i)21-s + (−1.11 − 1.11i)23-s + (0.960 + 0.279i)25-s + (0.136 + 0.136i)27-s − 0.288i·29-s − 0.479i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.412 + 0.911i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.412 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290757 - 0.450677i\)
\(L(\frac12)\) \(\approx\) \(0.290757 - 0.450677i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.315i)T \)
7 \( 1 + (-2.54 + 0.706i)T \)
good11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 + (1.47 - 1.47i)T - 13iT^{2} \)
17 \( 1 + (1.17 + 1.17i)T + 17iT^{2} \)
19 \( 1 + 2.09T + 19T^{2} \)
23 \( 1 + (5.33 + 5.33i)T + 23iT^{2} \)
29 \( 1 + 1.55iT - 29T^{2} \)
31 \( 1 + 2.66iT - 31T^{2} \)
37 \( 1 + (-0.549 + 0.549i)T - 37iT^{2} \)
41 \( 1 + 8.59iT - 41T^{2} \)
43 \( 1 + (5.86 + 5.86i)T + 43iT^{2} \)
47 \( 1 + (8.22 + 8.22i)T + 47iT^{2} \)
53 \( 1 + (-3.22 - 3.22i)T + 53iT^{2} \)
59 \( 1 + 3.54T + 59T^{2} \)
61 \( 1 + 4.08iT - 61T^{2} \)
67 \( 1 + (-3.06 + 3.06i)T - 67iT^{2} \)
71 \( 1 + 1.65T + 71T^{2} \)
73 \( 1 + (6.90 - 6.90i)T - 73iT^{2} \)
79 \( 1 + 12.6iT - 79T^{2} \)
83 \( 1 + (3.91 - 3.91i)T - 83iT^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (0.533 + 0.533i)T + 97iT^{2} \)
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

−10.20476824915215523597637637680, −8.946618883647464786153838032333, −8.241218992573956692159054582815, −7.45473992402066353792922313122, −6.53115903899405614562944247479, −5.24110924945651433714144074186, −4.51012686166930430004051415626, −3.78782158447303897214600817516, −2.16711614103122466680089113880, −0.27614216579845488982072952568, 1.57142391081683224519946664030, 2.98052213670058746894600526070, 4.30996713408608543512248336736, 5.09313190655047213180513377803, 6.13272972596589038146831996981, 7.19134489930116888533281304862, 8.001837122858006726279278381607, 8.368085295998125159594817178250, 9.719797629363897375559342247234, 10.71953903777705016938574593573

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