↑

Properties

Label 2-1088-136.29-c0-0-0
Degree $2$
Conductor $1088$
Sign $0.971 - 0.238i$
Analytic cond. $0.542982$
Root an. cond. $0.736873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.216 + 0.324i)3-s + (0.324 − 0.783i)9-s + (1.38 + 0.923i)11-s + (−0.382 − 0.923i)17-s + (−1.30 + 0.541i)19-s + (0.923 + 0.382i)25-s + (0.707 − 0.140i)27-s + 0.648i·33-s + (1.08 − 0.216i)41-s + (0.707 + 0.292i)43-s + (−0.923 + 0.382i)49-s + (0.216 − 0.324i)51-s + (−0.458 − 0.306i)57-s + (−0.292 + 0.707i)59-s − 1.84·67-s + ⋯
L(s)  = 1  + (0.216 + 0.324i)3-s + (0.324 − 0.783i)9-s + (1.38 + 0.923i)11-s + (−0.382 − 0.923i)17-s + (−1.30 + 0.541i)19-s + (0.923 + 0.382i)25-s + (0.707 − 0.140i)27-s + 0.648i·33-s + (1.08 − 0.216i)41-s + (0.707 + 0.292i)43-s + (−0.923 + 0.382i)49-s + (0.216 − 0.324i)51-s + (−0.458 − 0.306i)57-s + (−0.292 + 0.707i)59-s − 1.84·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.971 - 0.238i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :0),\ 0.971 - 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.190696188\)
\(L(\frac12)\) \(\approx\) \(1.190696188\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (0.382 + 0.923i)T \)
good3 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (-1.38 - 0.923i)T + (0.382 + 0.923i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.382 + 0.923i)T^{2} \)
29 \( 1 + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \)
43 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.923 - 0.382i)T^{2} \)
67 \( 1 + 1.84T + T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
97 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
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

−9.956291815457497059859031429842, −9.186037362368264182701139244294, −8.819876786258835174009316963797, −7.47879516525162669441868956171, −6.75986570332986159721761249538, −6.03584063979261360873884065801, −4.55725211037460455464943378410, −4.12579787185840971722806023753, −2.90711221637795392104348716922, −1.48797602761900290081246264541, 1.43757761804064074477524117233, 2.62678069301145085649344856313, 3.91781304425821782041347559518, 4.68918918315549368526915815466, 6.04855977818639413640854670626, 6.60255567083117778832111232831, 7.56798819259149340576043503441, 8.633373908275411256493182462891, 8.848868583966828763229917413918, 10.12615558211760764960619272131

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