↑

Properties

Label 2-42e2-28.27-c1-0-89
Degree $2$
Conductor $1764$
Sign $-0.780 + 0.624i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (1.20 − 0.736i)2-s + (0.914 − 1.77i)4-s − 1.08i·5-s + (−0.207 − 2.82i)8-s + (−0.797 − 1.30i)10-s − 2.08i·11-s − 2.61i·13-s + (−2.32 − 3.25i)16-s + 4.46i·17-s − 1.12·19-s + (−1.92 − 0.989i)20-s + (−1.53 − 2.51i)22-s − 7.11i·23-s + 3.82·25-s + (−1.92 − 3.15i)26-s + ⋯
L(s)  = 1  + (0.853 − 0.521i)2-s + (0.457 − 0.889i)4-s − 0.484i·5-s + (−0.0732 − 0.997i)8-s + (−0.252 − 0.413i)10-s − 0.628i·11-s − 0.724i·13-s + (−0.582 − 0.813i)16-s + 1.08i·17-s − 0.258·19-s + (−0.430 − 0.221i)20-s + (−0.327 − 0.536i)22-s − 1.48i·23-s + 0.765·25-s + (−0.377 − 0.618i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.780 + 0.624i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.780 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.496956445\)
\(L(\frac12)\) \(\approx\) \(2.496956445\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 + 0.736i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.08iT - 5T^{2} \)
11 \( 1 + 2.08iT - 11T^{2} \)
13 \( 1 + 2.61iT - 13T^{2} \)
17 \( 1 - 4.46iT - 17T^{2} \)
19 \( 1 + 1.12T + 19T^{2} \)
23 \( 1 + 7.11iT - 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
31 \( 1 + 7.70T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 5.54iT - 41T^{2} \)
43 \( 1 + 7.97iT - 43T^{2} \)
47 \( 1 - 5.44T + 47T^{2} \)
53 \( 1 - 6.48T + 53T^{2} \)
59 \( 1 + 8.82T + 59T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 - 10.0iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 7.97iT - 73T^{2} \)
79 \( 1 + 4.16iT - 79T^{2} \)
83 \( 1 - 4.31T + 83T^{2} \)
89 \( 1 + 4.01iT - 89T^{2} \)
97 \( 1 + 3.82iT - 97T^{2} \)
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

−8.900594852773484442860458183211, −8.412290206913716251824154403797, −7.23702513866643936883497666439, −6.31238968845515714084955142399, −5.63145996920713806617923685455, −4.82429440723239203213961808552, −3.93246217242978761109293089812, −3.07369086066665307742814317111, −1.95363382364094375570057042803, −0.67469276887324311426059706139, 1.88255628763068660203725018144, 2.93381265723579301954906721011, 3.83627355501274882655575733580, 4.75853754911695563826367509836, 5.50255968079596157509742654567, 6.45940339815753232088175858480, 7.25369868437197948223324846121, 7.54168044228543023987660427482, 8.844903837795277742719985682860, 9.400905341930517296740188666672

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