↑

Properties

Label 2-88-8.5-c1-0-1
Degree $2$
Conductor $88$
Sign $0.939 - 0.342i$
Analytic cond. $0.702683$
Root an. cond. $0.838262$
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  + (−1.29 − 0.559i)2-s + 0.229i·3-s + (1.37 + 1.45i)4-s + 2.51i·5-s + (0.128 − 0.298i)6-s + 1.47·7-s + (−0.968 − 2.65i)8-s + 2.94·9-s + (1.40 − 3.26i)10-s − i·11-s + (−0.334 + 0.315i)12-s + 3.47i·13-s + (−1.91 − 0.827i)14-s − 0.578·15-s + (−0.229 + 3.99i)16-s − 3.31·17-s + ⋯
L(s)  = 1  + (−0.918 − 0.395i)2-s + 0.132i·3-s + (0.686 + 0.727i)4-s + 1.12i·5-s + (0.0525 − 0.121i)6-s + 0.558·7-s + (−0.342 − 0.939i)8-s + 0.982·9-s + (0.445 − 1.03i)10-s − 0.301i·11-s + (−0.0965 + 0.0911i)12-s + 0.964i·13-s + (−0.512 − 0.221i)14-s − 0.149·15-s + (−0.0574 + 0.998i)16-s − 0.803·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.705704 + 0.124613i\)
\(L(\frac12)\) \(\approx\) \(0.705704 + 0.124613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.559i)T \)
11 \( 1 + iT \)
good3 \( 1 - 0.229iT - 3T^{2} \)
5 \( 1 - 2.51iT - 5T^{2} \)
7 \( 1 - 1.47T + 7T^{2} \)
13 \( 1 - 3.47iT - 13T^{2} \)
17 \( 1 + 3.31T + 17T^{2} \)
19 \( 1 + 7.13iT - 19T^{2} \)
23 \( 1 + 6.45T + 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 - 0.636T + 31T^{2} \)
37 \( 1 + 6.97iT - 37T^{2} \)
41 \( 1 - 6.72T + 41T^{2} \)
43 \( 1 + 3.21iT - 43T^{2} \)
47 \( 1 - 0.862T + 47T^{2} \)
53 \( 1 + 13.2iT - 53T^{2} \)
59 \( 1 - 2.63iT - 59T^{2} \)
61 \( 1 - 6.45iT - 61T^{2} \)
67 \( 1 - 7.66iT - 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + 1.04T + 89T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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

−14.31307252914243163428684492200, −13.06354976329468795140583913153, −11.58825532070636520517894590556, −10.97372030713300449313786530469, −9.949995054840283908091468535581, −8.830864629349378032646146020987, −7.38833015247126225066001983737, −6.60028025318969638620349749977, −4.13226600625348143977321811058, −2.28882760431796179690276909616, 1.54434841411163915097944386236, 4.63245183487018408617213339315, 6.03159492866471002842978842133, 7.65333536616132586903164394983, 8.358425167261979975348272236991, 9.635879577679162668215210586880, 10.54045931091026259880857971796, 12.00722827976579758575030838306, 12.88837257601807577444727347678, 14.30753803604604903905623069416

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