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Properties

Label 2-88-11.5-c3-0-1
Degree $2$
Conductor $88$
Sign $0.593 - 0.804i$
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  + (−6.31 − 4.58i)3-s + (−3.65 + 11.2i)5-s + (7.70 − 5.60i)7-s + (10.4 + 32.2i)9-s + (30.3 + 20.2i)11-s + (20.4 + 63.0i)13-s + (74.7 − 54.3i)15-s + (2.53 − 7.79i)17-s + (88.2 + 64.1i)19-s − 74.3·21-s − 196.·23-s + (−12.3 − 8.94i)25-s + (16.7 − 51.6i)27-s + (−129. + 94.2i)29-s + (−25.0 − 77.2i)31-s + ⋯
L(s)  = 1  + (−1.21 − 0.883i)3-s + (−0.327 + 1.00i)5-s + (0.416 − 0.302i)7-s + (0.388 + 1.19i)9-s + (0.831 + 0.555i)11-s + (0.437 + 1.34i)13-s + (1.28 − 0.935i)15-s + (0.0361 − 0.111i)17-s + (1.06 + 0.774i)19-s − 0.773·21-s − 1.77·23-s + (−0.0984 − 0.0715i)25-s + (0.119 − 0.368i)27-s + (−0.831 + 0.603i)29-s + (−0.145 − 0.447i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.593 - 0.804i$
Analytic conductor: \(5.19216\)
Root analytic conductor: \(2.27863\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :3/2),\ 0.593 - 0.804i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.785702 + 0.396936i\)
\(L(\frac12)\) \(\approx\) \(0.785702 + 0.396936i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-30.3 - 20.2i)T \)
good3 \( 1 + (6.31 + 4.58i)T + (8.34 + 25.6i)T^{2} \)
5 \( 1 + (3.65 - 11.2i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (-7.70 + 5.60i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (-20.4 - 63.0i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (-2.53 + 7.79i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (-88.2 - 64.1i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + 196.T + 1.21e4T^{2} \)
29 \( 1 + (129. - 94.2i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (25.0 + 77.2i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (-94.2 + 68.4i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (-289. - 210. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 106.T + 7.95e4T^{2} \)
47 \( 1 + (210. + 153. i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (-170. - 525. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-362. + 263. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-165. + 508. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + 675.T + 3.00e5T^{2} \)
71 \( 1 + (234. - 723. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (-500. + 363. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-179. - 551. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (279. - 860. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 - 537.T + 7.04e5T^{2} \)
97 \( 1 + (574. + 1.76e3i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−13.93469464206975913917451624459, −12.43401640847337198395599394641, −11.58221545034552542307845208039, −11.07106830137574377775899099552, −9.626395458355299617913501092966, −7.66586249133520523669984043522, −6.86239574205685842785740037396, −5.90482945814154979325360232441, −4.07032565993840082637888949857, −1.58520732918764687842800214781, 0.67177888232135022234644451197, 3.92965712545785050405217067342, 5.17065304321924476415656176275, 5.97457182627543252887001647065, 8.005837841994473481454375527349, 9.181917234297547190619128728307, 10.37600826859315128691663360786, 11.49368410280977848252274019201, 12.05962272666478631006938372438, 13.33540589237471289997594638340

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