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Properties

Label 2-88-11.5-c1-0-1
Degree $2$
Conductor $88$
Sign $0.944 - 0.329i$
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  + (1.30 + 0.951i)3-s + (0.190 − 0.587i)5-s + (−1.30 + 0.951i)7-s + (−0.118 − 0.363i)9-s + (1.23 + 3.07i)11-s + (−1.80 − 5.56i)13-s + (0.809 − 0.587i)15-s + (−0.572 + 1.76i)17-s + (−3.92 − 2.85i)19-s − 2.61·21-s − 4·23-s + (3.73 + 2.71i)25-s + (1.69 − 5.20i)27-s + (−5.92 + 4.30i)29-s + (0.336 + 1.03i)31-s + ⋯
L(s)  = 1  + (0.755 + 0.549i)3-s + (0.0854 − 0.262i)5-s + (−0.494 + 0.359i)7-s + (−0.0393 − 0.121i)9-s + (0.372 + 0.927i)11-s + (−0.501 − 1.54i)13-s + (0.208 − 0.151i)15-s + (−0.138 + 0.427i)17-s + (−0.900 − 0.654i)19-s − 0.571·21-s − 0.834·23-s + (0.747 + 0.542i)25-s + (0.325 − 1.00i)27-s + (−1.10 + 0.799i)29-s + (0.0605 + 0.186i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.944 - 0.329i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ 0.944 - 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12108 + 0.190218i\)
\(L(\frac12)\) \(\approx\) \(1.12108 + 0.190218i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 3.07i)T \)
good3 \( 1 + (-1.30 - 0.951i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.190 + 0.587i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.30 - 0.951i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.80 + 5.56i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.572 - 1.76i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.92 + 2.85i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (5.92 - 4.30i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.336 - 1.03i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.78 + 5.65i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.78 - 5.65i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.52T + 43T^{2} \)
47 \( 1 + (-8.54 - 6.20i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.190 - 0.587i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.92 + 1.40i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.572 - 1.76i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + (1.57 - 4.84i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.54 + 1.84i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.19 + 3.66i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.89 + 8.92i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + (3.80 + 11.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.63728589141201142017890207983, −12.97063683353763014490457277602, −12.44868248569036383095529520014, −10.75783256480899482792160194696, −9.625390709378512383913604151011, −8.912621926737496808017444110918, −7.57722725373452583130631703521, −5.95159987271859050498514638788, −4.31896050633542380036399026447, −2.78718946515323995664647819920, 2.34078975853603948679166977390, 4.05643551663732213738238481037, 6.18444278864990889391846223786, 7.28270838749890953004482416683, 8.497903143780048054960115534978, 9.549394276781670604683352469790, 10.90525576995625567920765414624, 12.06882060277819161099230416187, 13.36291278171600563506987517234, 14.00909561356033508879730161101

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