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Properties

Label 2-88-88.59-c4-0-21
Degree $2$
Conductor $88$
Sign $0.444 + 0.895i$
Analytic cond. $9.09655$
Root an. cond. $3.01604$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.23 − 2.35i)2-s + (1.44 − 4.44i)3-s + (4.94 + 15.2i)4-s + (−15.1 + 10.9i)6-s + (19.7 − 60.8i)8-s + (47.8 + 34.7i)9-s + (105. + 58.5i)11-s + 74.8·12-s + (−207. + 150. i)16-s + (91.2 − 66.3i)17-s + (−73.0 − 224. i)18-s + (211. − 649. i)19-s + (−204. − 438. i)22-s + (−242. − 175. i)24-s + (193. − 594. i)25-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.160 − 0.494i)3-s + (0.309 + 0.951i)4-s + (−0.420 + 0.305i)6-s + (0.309 − 0.951i)8-s + (0.590 + 0.429i)9-s + (0.874 + 0.484i)11-s + 0.519·12-s + (−0.809 + 0.587i)16-s + (0.315 − 0.229i)17-s + (−0.225 − 0.694i)18-s + (0.584 − 1.79i)19-s + (−0.423 − 0.905i)22-s + (−0.420 − 0.305i)24-s + (0.309 − 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(9.09655\)
Root analytic conductor: \(3.01604\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :2),\ 0.444 + 0.895i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.12194 - 0.695572i\)
\(L(\frac12)\) \(\approx\) \(1.12194 - 0.695572i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.23 + 2.35i)T \)
11 \( 1 + (-105. - 58.5i)T \)
good3 \( 1 + (-1.44 + 4.44i)T + (-65.5 - 47.6i)T^{2} \)
5 \( 1 + (-193. + 594. i)T^{2} \)
7 \( 1 + (1.94e3 - 1.41e3i)T^{2} \)
13 \( 1 + (-8.82e3 - 2.71e4i)T^{2} \)
17 \( 1 + (-91.2 + 66.3i)T + (2.58e4 - 7.94e4i)T^{2} \)
19 \( 1 + (-211. + 649. i)T + (-1.05e5 - 7.66e4i)T^{2} \)
23 \( 1 - 2.79e5T^{2} \)
29 \( 1 + (5.72e5 - 4.15e5i)T^{2} \)
31 \( 1 + (-2.85e5 - 8.78e5i)T^{2} \)
37 \( 1 + (1.51e6 - 1.10e6i)T^{2} \)
41 \( 1 + (255. - 786. i)T + (-2.28e6 - 1.66e6i)T^{2} \)
43 \( 1 - 3.53e3T + 3.41e6T^{2} \)
47 \( 1 + (3.94e6 + 2.86e6i)T^{2} \)
53 \( 1 + (-2.43e6 - 7.50e6i)T^{2} \)
59 \( 1 + (-2.02e3 - 6.22e3i)T + (-9.80e6 + 7.12e6i)T^{2} \)
61 \( 1 + (-4.27e6 + 1.31e7i)T^{2} \)
67 \( 1 - 5.41e3T + 2.01e7T^{2} \)
71 \( 1 + (-7.85e6 + 2.41e7i)T^{2} \)
73 \( 1 + (1.50e3 + 4.61e3i)T + (-2.29e7 + 1.66e7i)T^{2} \)
79 \( 1 + (-1.20e7 - 3.70e7i)T^{2} \)
83 \( 1 + (-3.49e3 + 2.54e3i)T + (1.46e7 - 4.51e7i)T^{2} \)
89 \( 1 + 1.31e4T + 6.27e7T^{2} \)
97 \( 1 + (1.41e4 + 1.02e4i)T + (2.73e7 + 8.41e7i)T^{2} \)
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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

−12.99382850491571105694017583694, −12.10217356463847346891784556609, −11.06268037577707001340556432750, −9.864051914801854324407577991663, −8.900882901518331973382390914391, −7.58902957783936508892913029795, −6.75158363742265682633507032675, −4.42856246025513871552177196540, −2.58472349941362827910610810549, −1.05022637754391944793550538935, 1.29001454012665826923562734568, 3.77218264067010004188318746464, 5.56068200471182228632845433276, 6.80265977882379249957837622945, 8.078328195695941414437122025607, 9.264602314564812913596839337225, 9.997944441173116965609045737167, 11.15998407314981367659293911488, 12.43651483011718298484199764021, 14.11870766082066572119861634492

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