↑

Properties

Label 2-1323-63.59-c1-0-18
Degree $2$
Conductor $1323$
Sign $-0.761 + 0.648i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
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  + (−1.28 − 0.742i)2-s + (0.101 + 0.176i)4-s + 0.308·5-s + 2.66i·8-s + (−0.396 − 0.228i)10-s + 3.16i·11-s + (−3.00 − 1.73i)13-s + (2.18 − 3.78i)16-s + (−2.44 + 4.22i)17-s + (4.62 − 2.67i)19-s + (0.0314 + 0.0544i)20-s + (2.34 − 4.06i)22-s − 5.97i·23-s − 4.90·25-s + (2.57 + 4.45i)26-s + ⋯
L(s)  = 1  + (−0.909 − 0.524i)2-s + (0.0509 + 0.0882i)4-s + 0.137·5-s + 0.942i·8-s + (−0.125 − 0.0723i)10-s + 0.953i·11-s + (−0.833 − 0.481i)13-s + (0.545 − 0.945i)16-s + (−0.592 + 1.02i)17-s + (1.06 − 0.612i)19-s + (0.00702 + 0.0121i)20-s + (0.500 − 0.866i)22-s − 1.24i·23-s − 0.980·25-s + (0.504 + 0.874i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.761 + 0.648i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.761 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5556437011\)
\(L(\frac12)\) \(\approx\) \(0.5556437011\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (1.28 + 0.742i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - 0.308T + 5T^{2} \)
11 \( 1 - 3.16iT - 11T^{2} \)
13 \( 1 + (3.00 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.44 - 4.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.62 + 2.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.97iT - 23T^{2} \)
29 \( 1 + (2.70 - 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.51 + 3.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.92 + 10.2i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.58 + 4.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.75 - 4.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.23 + 7.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0740 - 0.0427i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.04 + 1.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.69 - 2.71i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0554 - 0.0959i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.78iT - 71T^{2} \)
73 \( 1 + (8.32 + 4.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.56 - 4.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.42 + 7.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.936 + 1.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.9 + 6.34i)T + (48.5 - 84.0i)T^{2} \)
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

−9.386964553810727934080963560188, −8.758621038979128282525128503032, −7.81025883329483297007221194875, −7.15492737312001065782424592963, −5.94611425796372780595863365751, −5.08570850856157530138666798292, −4.14910730871826072074495466913, −2.61519369326626195699149199592, −1.87370700327208694708376334309, −0.34540674927275815383760029645, 1.20217928469180933828155706470, 2.84086452069398113130874794138, 3.85214187714981689372258702142, 5.02889699587611442739857422120, 5.99969650842508018252708312871, 6.93118082619482154589743483718, 7.59625150056540241078965508818, 8.282698599109796673309549918125, 9.207182317681982353162924506565, 9.645535678312160263110621168802

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