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Properties

Label 2-88-88.35-c1-0-0
Degree $2$
Conductor $88$
Sign $-0.992 + 0.118i$
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  + (−1.39 + 0.236i)2-s + (−1.63 + 1.18i)3-s + (1.88 − 0.659i)4-s + (−1.62 + 0.527i)5-s + (1.99 − 2.04i)6-s + (−3.70 − 2.68i)7-s + (−2.47 + 1.36i)8-s + (0.333 − 1.02i)9-s + (2.13 − 1.11i)10-s + (−2.77 + 1.81i)11-s + (−2.30 + 3.31i)12-s + (−0.147 + 0.455i)13-s + (5.79 + 2.87i)14-s + (2.02 − 2.79i)15-s + (3.13 − 2.48i)16-s + (2.14 − 0.696i)17-s + ⋯
L(s)  = 1  + (−0.985 + 0.167i)2-s + (−0.943 + 0.685i)3-s + (0.944 − 0.329i)4-s + (−0.726 + 0.236i)5-s + (0.815 − 0.833i)6-s + (−1.39 − 1.01i)7-s + (−0.875 + 0.482i)8-s + (0.111 − 0.342i)9-s + (0.676 − 0.354i)10-s + (−0.836 + 0.547i)11-s + (−0.664 + 0.958i)12-s + (−0.0410 + 0.126i)13-s + (1.54 + 0.768i)14-s + (0.523 − 0.720i)15-s + (0.782 − 0.622i)16-s + (0.519 − 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.992 + 0.118i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ -0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00602894 - 0.101452i\)
\(L(\frac12)\) \(\approx\) \(0.00602894 - 0.101452i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.236i)T \)
11 \( 1 + (2.77 - 1.81i)T \)
good3 \( 1 + (1.63 - 1.18i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (1.62 - 0.527i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.70 + 2.68i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.147 - 0.455i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.14 + 0.696i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.24 - 1.70i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 7.45iT - 23T^{2} \)
29 \( 1 + (3.30 + 2.40i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.84 - 1.24i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.89 - 5.36i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.72 + 5.12i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.32iT - 43T^{2} \)
47 \( 1 + (2.59 + 3.56i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.67 - 0.545i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.30 + 4.58i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.782 + 2.40i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.41T + 67T^{2} \)
71 \( 1 + (-4.63 + 1.50i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.11 - 5.65i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (5.15 - 15.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.14 + 1.67i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 + (-1.68 + 5.17i)T + (-78.4 - 57.0i)T^{2} \)
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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

−15.59329129771932502879905883207, −13.65539236217386775302325741965, −12.16311861573743826315120318889, −11.19890860207561860371705794236, −10.16624078646307797970287284484, −9.749806602964571536271730336836, −7.79840939998768014017383862869, −6.91150016766881664460612607662, −5.49234344182760823629087466078, −3.56099422840583305787228876811, 0.17758857340468193624863782384, 2.98995671515891826617308383038, 5.77241550085864671789813430626, 6.66841344334401924999528759757, 7.990279029781178655956386942381, 9.128282723154395896303680337306, 10.40786900982424198144374060084, 11.58040719919669759756004441362, 12.36072938306015176598042246779, 12.93530760850821279677898391125

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