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Properties

Label 2-88-11.4-c1-0-2
Degree $2$
Conductor $88$
Sign $0.451 + 0.892i$
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  + (0.941 − 2.89i)3-s + (−1.44 + 1.04i)5-s + (0.109 + 0.335i)7-s + (−5.08 − 3.69i)9-s + (2.91 + 1.58i)11-s + (4.48 + 3.26i)13-s + (1.67 + 5.16i)15-s + (−3.08 + 2.24i)17-s + (1.42 − 4.39i)19-s + 1.07·21-s − 7.00·23-s + (−0.564 + 1.73i)25-s + (−8.09 + 5.87i)27-s + (0.654 + 2.01i)29-s + (−2.79 − 2.03i)31-s + ⋯
L(s)  = 1  + (0.543 − 1.67i)3-s + (−0.644 + 0.468i)5-s + (0.0412 + 0.126i)7-s + (−1.69 − 1.23i)9-s + (0.878 + 0.477i)11-s + (1.24 + 0.904i)13-s + (0.433 + 1.33i)15-s + (−0.748 + 0.543i)17-s + (0.327 − 1.00i)19-s + 0.234·21-s − 1.46·23-s + (−0.112 + 0.347i)25-s + (−1.55 + 1.13i)27-s + (0.121 + 0.374i)29-s + (−0.501 − 0.364i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.451 + 0.892i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ 0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.897234 - 0.551475i\)
\(L(\frac12)\) \(\approx\) \(0.897234 - 0.551475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-2.91 - 1.58i)T \)
good3 \( 1 + (-0.941 + 2.89i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.44 - 1.04i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.109 - 0.335i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.48 - 3.26i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.08 - 2.24i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.42 + 4.39i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 7.00T + 23T^{2} \)
29 \( 1 + (-0.654 - 2.01i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.79 + 2.03i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.537 + 1.65i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.458 - 1.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.92T + 43T^{2} \)
47 \( 1 + (0.374 - 1.15i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.41 + 4.65i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.954 + 2.93i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.15 + 4.47i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + (-1.91 + 1.39i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.20 - 3.71i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.32 + 2.41i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.662 - 0.481i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.92T + 89T^{2} \)
97 \( 1 + (-7.66 - 5.57i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.87329914504623890770207480852, −13.02461726912958262388515347846, −11.88027918520461250141431876752, −11.24373544012049522233079305176, −9.155914966356329772932417887961, −8.200496134729543668224969288717, −7.03839220304127651942640876292, −6.34556284675936029893112443808, −3.75485069214614195799214868475, −1.87329240694976561342095209592, 3.50160437216349931700718841341, 4.30769204199739065159331135535, 5.84078559867491675572117746424, 8.114665337338121872337319825730, 8.808940991412169422260438003581, 9.964120166991195481853827369478, 10.94669178122728207426639921973, 11.97033997093902836786520546942, 13.65189736646932778630139028396, 14.46733709110506388573490812997

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