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Properties

Label 2-88-8.5-c5-0-7
Degree $2$
Conductor $88$
Sign $-0.361 + 0.932i$
Analytic cond. $14.1137$
Root an. cond. $3.75683$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.20 + 5.20i)2-s + 27.5i·3-s + (−22.2 + 22.9i)4-s − 11.7i·5-s + (−143. + 60.7i)6-s + 48.1·7-s + (−168. − 65.3i)8-s − 515.·9-s + (61.1 − 25.8i)10-s + 121i·11-s + (−632. − 613. i)12-s + 425. i·13-s + (106. + 250. i)14-s + 323.·15-s + (−31.5 − 1.02e3i)16-s − 350.·17-s + ⋯
L(s)  = 1  + (0.389 + 0.920i)2-s + 1.76i·3-s + (−0.696 + 0.717i)4-s − 0.210i·5-s + (−1.62 + 0.688i)6-s + 0.371·7-s + (−0.932 − 0.361i)8-s − 2.12·9-s + (0.193 − 0.0818i)10-s + 0.301i·11-s + (−1.26 − 1.23i)12-s + 0.697i·13-s + (0.144 + 0.341i)14-s + 0.371·15-s + (−0.0307 − 0.999i)16-s − 0.294·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.361 + 0.932i$
Analytic conductor: \(14.1137\)
Root analytic conductor: \(3.75683\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :5/2),\ -0.361 + 0.932i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.818394 - 1.19472i\)
\(L(\frac12)\) \(\approx\) \(0.818394 - 1.19472i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.20 - 5.20i)T \)
11 \( 1 - 121iT \)
good3 \( 1 - 27.5iT - 243T^{2} \)
5 \( 1 + 11.7iT - 3.12e3T^{2} \)
7 \( 1 - 48.1T + 1.68e4T^{2} \)
13 \( 1 - 425. iT - 3.71e5T^{2} \)
17 \( 1 + 350.T + 1.41e6T^{2} \)
19 \( 1 + 430. iT - 2.47e6T^{2} \)
23 \( 1 - 3.42e3T + 6.43e6T^{2} \)
29 \( 1 - 4.61e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.30e3T + 2.86e7T^{2} \)
37 \( 1 - 8.21e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.64e4T + 1.15e8T^{2} \)
43 \( 1 + 1.01e4iT - 1.47e8T^{2} \)
47 \( 1 - 4.04e3T + 2.29e8T^{2} \)
53 \( 1 - 2.61e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.35e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.72e3iT - 8.44e8T^{2} \)
67 \( 1 - 6.83e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.57e4T + 1.80e9T^{2} \)
73 \( 1 - 4.71e4T + 2.07e9T^{2} \)
79 \( 1 + 3.23e3T + 3.07e9T^{2} \)
83 \( 1 + 6.02e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.09e5T + 5.58e9T^{2} \)
97 \( 1 - 5.26e4T + 8.58e9T^{2} \)
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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

−14.49151076325621722992252110405, −13.15259410703767840137477897069, −11.69116978748271645317951811963, −10.55891445369595466527763599455, −9.215318630230035828809043250091, −8.685820087651049162235099444842, −6.94072084131608286526782017979, −5.25186466373006272134229659281, −4.61944061751518233662246358143, −3.35339345664801309977388577842, 0.52675772666420903314749893682, 1.80853226349852482055710034915, 3.11125314291452845490328484022, 5.29258411872094640555611519756, 6.53130958649607503058736419047, 7.84951005223436789125385635958, 8.991950295431725930323425750219, 10.76651033306834427226380185414, 11.54977375905089510774978532824, 12.65764893626618615976322112538

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