↑

Properties

Label 2-12e3-9.5-c2-0-22
Degree $2$
Conductor $1728$
Sign $-0.186 - 0.982i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (6.92 + 3.99i)5-s + (6.10 + 10.5i)7-s + (2.04 − 1.17i)11-s + (−8.01 + 13.8i)13-s − 9.26i·17-s + 4.43·19-s + (10.9 + 6.29i)23-s + (19.4 + 33.7i)25-s + (21.3 − 12.3i)29-s + (−17.9 + 31.1i)31-s + 97.6i·35-s + 62.3·37-s + (−19.9 − 11.5i)41-s + (18.4 + 31.9i)43-s + (−18.4 + 10.6i)47-s + ⋯
L(s)  = 1  + (1.38 + 0.799i)5-s + (0.872 + 1.51i)7-s + (0.185 − 0.107i)11-s + (−0.616 + 1.06i)13-s − 0.544i·17-s + 0.233·19-s + (0.474 + 0.273i)23-s + (0.778 + 1.34i)25-s + (0.737 − 0.425i)29-s + (−0.579 + 1.00i)31-s + 2.79i·35-s + 1.68·37-s + (−0.487 − 0.281i)41-s + (0.429 + 0.743i)43-s + (−0.392 + 0.226i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.186 - 0.982i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.186 - 0.982i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.965335563\)
\(L(\frac12)\) \(\approx\) \(2.965335563\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-6.92 - 3.99i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-6.10 - 10.5i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-2.04 + 1.17i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (8.01 - 13.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 9.26iT - 289T^{2} \)
19 \( 1 - 4.43T + 361T^{2} \)
23 \( 1 + (-10.9 - 6.29i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-21.3 + 12.3i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (17.9 - 31.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 62.3T + 1.36e3T^{2} \)
41 \( 1 + (19.9 + 11.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-18.4 - 31.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (18.4 - 10.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 94.9iT - 2.80e3T^{2} \)
59 \( 1 + (46.4 + 26.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (53.0 + 91.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (5.50 - 9.52i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 108. iT - 5.04e3T^{2} \)
73 \( 1 - 26.0T + 5.32e3T^{2} \)
79 \( 1 + (26.7 + 46.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-111. + 64.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 39.8iT - 7.92e3T^{2} \)
97 \( 1 + (29.4 + 51.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−9.415733637658061812836534289036, −8.789058493565040528683532959136, −7.78728553274200071008873693495, −6.74729887234912883053908215294, −6.17392936873085585792212531570, −5.31143418811349857246836873542, −4.71741526482841110416673292994, −3.04795500628336250126797762240, −2.30529256588215377301671697246, −1.60467934911815542283770985014, 0.790407985184669405206191066664, 1.50108953262005004873689365717, 2.69219301703518623714648095248, 4.11509818688321474528481378146, 4.82489997461599166119948188502, 5.56346987744190616617776722851, 6.42839225411885137393145504707, 7.50118400162084925721583570039, 8.008422806922478095257524236451, 9.042287364326168173314431093316

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