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Properties

Label 2-88-88.19-c1-0-6
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

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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: $\chi_{88} (19, \cdot )$
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 \( 1 + (-5.12 + 3.72i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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