↑

Properties

Label 2-88-88.51-c1-0-2
Degree $2$
Conductor $88$
Sign $0.851 - 0.524i$
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  + (−0.557 − 1.29i)2-s + (−0.625 + 1.92i)3-s + (−1.37 + 1.44i)4-s + (1.75 + 2.41i)5-s + (2.85 − 0.259i)6-s + (−0.216 − 0.667i)7-s + (2.65 + 0.986i)8-s + (−0.888 − 0.645i)9-s + (2.15 − 3.61i)10-s + (−2.75 − 1.85i)11-s + (−1.92 − 3.56i)12-s + (2.50 + 1.82i)13-s + (−0.746 + 0.653i)14-s + (−5.73 + 1.86i)15-s + (−0.194 − 3.99i)16-s + (−2.54 − 3.49i)17-s + ⋯
L(s)  = 1  + (−0.393 − 0.919i)2-s + (−0.361 + 1.11i)3-s + (−0.689 + 0.724i)4-s + (0.783 + 1.07i)5-s + (1.16 − 0.105i)6-s + (−0.0819 − 0.252i)7-s + (0.937 + 0.348i)8-s + (−0.296 − 0.215i)9-s + (0.682 − 1.14i)10-s + (−0.829 − 0.558i)11-s + (−0.555 − 1.02i)12-s + (0.695 + 0.505i)13-s + (−0.199 + 0.174i)14-s + (−1.48 + 0.481i)15-s + (−0.0485 − 0.998i)16-s + (−0.616 − 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.851 - 0.524i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ 0.851 - 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.734369 + 0.207983i\)
\(L(\frac12)\) \(\approx\) \(0.734369 + 0.207983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.557 + 1.29i)T \)
11 \( 1 + (2.75 + 1.85i)T \)
good3 \( 1 + (0.625 - 1.92i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.75 - 2.41i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.216 + 0.667i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.50 - 1.82i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.54 + 3.49i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.99 - 1.62i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.883iT - 23T^{2} \)
29 \( 1 + (3.13 + 9.66i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.71 + 6.48i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.84 + 0.924i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.79 - 0.906i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.57iT - 43T^{2} \)
47 \( 1 + (-4.57 - 1.48i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.22 - 9.94i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.63 + 5.02i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (6.21 - 4.51i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.11T + 67T^{2} \)
71 \( 1 + (-4.66 - 6.42i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.05 - 1.31i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.28 + 1.65i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.679 + 0.934i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.59T + 89T^{2} \)
97 \( 1 + (-5.12 - 3.72i)T + (29.9 + 92.2i)T^{2} \)
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

−13.88766075770936174101193652609, −13.43588995351198205038943434712, −11.51035268666345554226788954314, −10.92980736035433587222696580603, −10.00677895729944696280812130971, −9.386885288759884905016216392611, −7.63922080817994057213989635407, −5.82375903364292483226871385229, −4.22283467582165721369653093851, −2.74549841238561291616282349033, 1.39132359162629863650237443729, 5.04134714166285434628475990148, 5.97154566255157912177152451636, 7.15858311901059017904266306557, 8.342140568076598195761721452756, 9.329275614361902517064705372097, 10.62538482912218050432596206466, 12.48676481114514942933529671509, 13.04930462799050854760384395600, 13.83207049000598402774789924526

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