↑

Properties

Label 2-88-88.43-c3-0-11
Degree $2$
Conductor $88$
Sign $0.912 - 0.409i$
Analytic cond. $5.19216$
Root an. cond. $2.27863$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.60 − 1.10i)2-s − 7.48·3-s + (5.55 − 5.75i)4-s + 19.6i·5-s + (−19.4 + 8.28i)6-s + 30.5·7-s + (8.08 − 21.1i)8-s + 29.0·9-s + (21.7 + 51.1i)10-s + (25.8 + 25.7i)11-s + (−41.5 + 43.1i)12-s − 18.4·13-s + (79.4 − 33.7i)14-s − 147. i·15-s + (−2.33 − 63.9i)16-s + 68.1i·17-s + ⋯
L(s)  = 1  + (0.920 − 0.391i)2-s − 1.44·3-s + (0.694 − 0.719i)4-s + 1.75i·5-s + (−1.32 + 0.563i)6-s + 1.64·7-s + (0.357 − 0.934i)8-s + 1.07·9-s + (0.686 + 1.61i)10-s + (0.708 + 0.706i)11-s + (−1.00 + 1.03i)12-s − 0.394·13-s + (1.51 − 0.644i)14-s − 2.53i·15-s + (−0.0364 − 0.999i)16-s + 0.971i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.912 - 0.409i$
Analytic conductor: \(5.19216\)
Root analytic conductor: \(2.27863\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :3/2),\ 0.912 - 0.409i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.86488 + 0.399031i\)
\(L(\frac12)\) \(\approx\) \(1.86488 + 0.399031i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.60 + 1.10i)T \)
11 \( 1 + (-25.8 - 25.7i)T \)
good3 \( 1 + 7.48T + 27T^{2} \)
5 \( 1 - 19.6iT - 125T^{2} \)
7 \( 1 - 30.5T + 343T^{2} \)
13 \( 1 + 18.4T + 2.19e3T^{2} \)
17 \( 1 - 68.1iT - 4.91e3T^{2} \)
19 \( 1 - 11.0iT - 6.85e3T^{2} \)
23 \( 1 - 37.9iT - 1.21e4T^{2} \)
29 \( 1 - 19.6T + 2.43e4T^{2} \)
31 \( 1 + 126. iT - 2.97e4T^{2} \)
37 \( 1 + 81.2iT - 5.06e4T^{2} \)
41 \( 1 + 41.9iT - 6.89e4T^{2} \)
43 \( 1 + 400. iT - 7.95e4T^{2} \)
47 \( 1 + 265. iT - 1.03e5T^{2} \)
53 \( 1 + 218. iT - 1.48e5T^{2} \)
59 \( 1 + 123.T + 2.05e5T^{2} \)
61 \( 1 + 196.T + 2.26e5T^{2} \)
67 \( 1 + 619.T + 3.00e5T^{2} \)
71 \( 1 + 518. iT - 3.57e5T^{2} \)
73 \( 1 - 1.18e3iT - 3.89e5T^{2} \)
79 \( 1 - 1.06e3T + 4.93e5T^{2} \)
83 \( 1 - 725. iT - 5.71e5T^{2} \)
89 \( 1 - 857.T + 7.04e5T^{2} \)
97 \( 1 + 184.T + 9.12e5T^{2} \)
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

−13.93414290927851352239168570182, −12.22559794255576611728718149682, −11.53395288118572941052085761971, −10.88717913121130651314134926254, −10.18091518386663822580136905476, −7.43516622570926012847406935431, −6.49043467773709444344044000694, −5.39627678309207052654431572046, −4.08044843064108832416200327296, −1.93048111064626057055777742069, 1.19619084612833284210065073956, 4.67485773213684224073411105378, 4.93265562883606058208853303005, 6.08735476444308183279540108385, 7.75174247047281019220959882383, 8.880103859672236077463116065498, 10.99168267874379856997807484831, 11.80896752064372060114384300869, 12.24689287354758940494108015567, 13.50442377752166566942346468636

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