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Properties

Label 2-88-11.4-c3-0-2
Degree $2$
Conductor $88$
Sign $0.320 - 0.947i$
Analytic cond. $5.19216$
Root an. cond. $2.27863$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.320 − 0.986i)3-s + (−7.72 + 5.61i)5-s + (5.56 + 17.1i)7-s + (20.9 + 15.2i)9-s + (10.0 + 35.0i)11-s + (−13.3 − 9.68i)13-s + (3.05 + 9.41i)15-s + (64.6 − 46.9i)17-s + (−44.2 + 136. i)19-s + 18.6·21-s − 45.6·23-s + (−10.4 + 32.2i)25-s + (44.4 − 32.2i)27-s + (−45.4 − 140. i)29-s + (12.6 + 9.17i)31-s + ⋯
L(s)  = 1  + (0.0616 − 0.189i)3-s + (−0.690 + 0.501i)5-s + (0.300 + 0.924i)7-s + (0.776 + 0.564i)9-s + (0.274 + 0.961i)11-s + (−0.284 − 0.206i)13-s + (0.0526 + 0.162i)15-s + (0.921 − 0.669i)17-s + (−0.534 + 1.64i)19-s + 0.194·21-s − 0.413·23-s + (−0.0837 + 0.257i)25-s + (0.316 − 0.229i)27-s + (−0.291 − 0.896i)29-s + (0.0731 + 0.0531i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(5.19216\)
Root analytic conductor: \(2.27863\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :3/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.11146 + 0.797433i\)
\(L(\frac12)\) \(\approx\) \(1.11146 + 0.797433i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-10.0 - 35.0i)T \)
good3 \( 1 + (-0.320 + 0.986i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (7.72 - 5.61i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (-5.56 - 17.1i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (13.3 + 9.68i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-64.6 + 46.9i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (44.2 - 136. i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + 45.6T + 1.21e4T^{2} \)
29 \( 1 + (45.4 + 140. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-12.6 - 9.17i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (89.0 + 273. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-92.6 + 285. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 - 125.T + 7.95e4T^{2} \)
47 \( 1 + (9.79 - 30.1i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (-421. - 306. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (70.9 + 218. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (600. - 436. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 - 505.T + 3.00e5T^{2} \)
71 \( 1 + (-689. + 500. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (-74.0 - 227. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-592. - 430. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (476. - 346. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 - 663.T + 7.04e5T^{2} \)
97 \( 1 + (-1.10e3 - 801. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−14.04971802855791245037696824608, −12.46006583828401738468480739157, −12.03409501621509028252765340579, −10.61894685065999531509403799743, −9.558233600976919367126902527861, −7.981100697439189103698221514180, −7.25881705224679824438559409423, −5.57917599008207211060214571201, −4.01378462793041007743419663751, −2.09724432465332709856840835441, 0.884404961763795539274292700933, 3.64969807218502900969894236056, 4.69862251548027173648108812646, 6.59216736086487050842810710852, 7.81708269147632165910779010020, 8.934530276723199209495809154380, 10.23565643109568536564366702518, 11.30582785588609036753635064385, 12.38854354767880553454020057521, 13.43566945201900483762710771181

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