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Properties

Label 2-88-88.19-c1-0-0
Degree $2$
Conductor $88$
Sign $0.519 - 0.854i$
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  + (−1.16 − 0.806i)2-s + (0.385 + 1.18i)3-s + (0.699 + 1.87i)4-s + (−2.03 + 2.80i)5-s + (0.509 − 1.69i)6-s + (−0.442 + 1.36i)7-s + (0.698 − 2.74i)8-s + (1.16 − 0.847i)9-s + (4.62 − 1.61i)10-s + (−1.18 + 3.09i)11-s + (−1.95 + 1.55i)12-s + (5.33 − 3.87i)13-s + (1.61 − 1.22i)14-s + (−4.10 − 1.33i)15-s + (−3.02 + 2.62i)16-s + (0.442 − 0.609i)17-s + ⋯
L(s)  = 1  + (−0.821 − 0.570i)2-s + (0.222 + 0.685i)3-s + (0.349 + 0.936i)4-s + (−0.909 + 1.25i)5-s + (0.207 − 0.689i)6-s + (−0.167 + 0.514i)7-s + (0.247 − 0.969i)8-s + (0.388 − 0.282i)9-s + (1.46 − 0.509i)10-s + (−0.357 + 0.933i)11-s + (−0.564 + 0.448i)12-s + (1.48 − 1.07i)13-s + (0.430 − 0.327i)14-s + (−1.06 − 0.344i)15-s + (−0.755 + 0.655i)16-s + (0.107 − 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.519 - 0.854i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ 0.519 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567402 + 0.319130i\)
\(L(\frac12)\) \(\approx\) \(0.567402 + 0.319130i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.806i)T \)
11 \( 1 + (1.18 - 3.09i)T \)
good3 \( 1 + (-0.385 - 1.18i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.03 - 2.80i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.442 - 1.36i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-5.33 + 3.87i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.442 + 0.609i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.61 - 0.850i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.40iT - 23T^{2} \)
29 \( 1 + (0.0386 - 0.119i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.25 - 3.10i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.91 - 0.622i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.52 + 0.818i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.68iT - 43T^{2} \)
47 \( 1 + (-4.39 + 1.42i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.06 + 4.21i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.90 + 5.86i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.37 + 1.72i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + (6.91 - 9.51i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.95 + 1.93i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.32 - 0.963i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.17 + 9.88i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + (9.17 - 6.66i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.77564806652124877316646658861, −12.94277102095830221392618591347, −11.96963919360380701675715454935, −10.65894514823073775519234198307, −10.34161152537450232534668416545, −8.900026920208764247982156588598, −7.79177007048150637751354477331, −6.59031923716478150634849138299, −4.01928237716086594159399704381, −2.92704452867224781897427319544, 1.18948333943983729733875763743, 4.31661286973348634525839953656, 6.09034181989652474429487368301, 7.46793539757118589967276951407, 8.304083507174555341753904610780, 9.091634886984998307262631636597, 10.74300460991320560894098926967, 11.73910438018282399074163668784, 13.19725985013648471868946716312, 13.78620950102033416729805906170

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