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Properties

Label 2-88-88.51-c1-0-9
Degree $2$
Conductor $88$
Sign $0.419 + 0.907i$
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  + (0.831 − 1.14i)2-s + (0.0137 − 0.0422i)3-s + (−0.618 − 1.90i)4-s + (−0.0369 − 0.0508i)6-s + (−2.68 − 0.874i)8-s + (2.42 + 1.76i)9-s + (−2.27 + 2.41i)11-s − 0.0889·12-s + (−3.23 + 2.35i)16-s + (2.32 + 3.20i)17-s + (4.03 − 1.31i)18-s + (−4.30 − 1.39i)19-s + (0.875 + 4.60i)22-s + (−0.0739 + 0.101i)24-s + (1.54 − 4.75i)25-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (0.00793 − 0.0244i)3-s + (−0.309 − 0.951i)4-s + (−0.0150 − 0.0207i)6-s + (−0.951 − 0.309i)8-s + (0.808 + 0.587i)9-s + (−0.685 + 0.728i)11-s − 0.0256·12-s + (−0.809 + 0.587i)16-s + (0.564 + 0.776i)17-s + (0.950 − 0.308i)18-s + (−0.987 − 0.320i)19-s + (0.186 + 0.982i)22-s + (−0.0150 + 0.0207i)24-s + (0.309 − 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.419 + 0.907i$
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: \((2,\ 88,\ (\ :1/2),\ 0.419 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03241 - 0.660553i\)
\(L(\frac12)\) \(\approx\) \(1.03241 - 0.660553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 + 1.14i)T \)
11 \( 1 + (2.27 - 2.41i)T \)
good3 \( 1 + (-0.0137 + 0.0422i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.32 - 3.20i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.30 + 1.39i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (12.0 + 3.91i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 12.7iT - 43T^{2} \)
47 \( 1 + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.58 - 11.0i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (11.9 - 3.87i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.56 - 10.4i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 + (-14.6 - 10.6i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.70486813583762362750490804096, −12.85016479624564582342847129064, −12.07522051875904849237077620884, −10.54026397078329084441583086334, −10.14959544655051385948700930925, −8.518931536449895639772158029593, −6.91825099529574461439471541086, −5.30373562686937902616021038939, −4.09575242349162953591860292632, −2.15046371750999438062685557782, 3.33760854451402009910609474374, 4.85999825168689742801736587267, 6.20537618382646995818090549253, 7.38372240307539742687941200420, 8.540734167591234418640306691457, 9.839417699290562186095771863436, 11.36880076804928764360862301749, 12.59676458021694954121804475811, 13.32419661963486155674511511411, 14.46559950469378267398969277567

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