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Properties

Label 2-88-88.5-c1-0-0
Degree $2$
Conductor $88$
Sign $-0.495 - 0.868i$
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  + (−0.568 − 1.29i)2-s + (−1.50 + 2.06i)3-s + (−1.35 + 1.47i)4-s + (−2.56 − 0.832i)5-s + (3.53 + 0.769i)6-s + (−3.19 + 2.32i)7-s + (2.67 + 0.915i)8-s + (−1.08 − 3.35i)9-s + (0.379 + 3.79i)10-s + (3.20 − 0.869i)11-s + (−1.01 − 5.00i)12-s + (−0.416 + 0.135i)13-s + (4.82 + 2.81i)14-s + (5.57 − 4.04i)15-s + (−0.336 − 3.98i)16-s + (−0.739 + 2.27i)17-s + ⋯
L(s)  = 1  + (−0.402 − 0.915i)2-s + (−0.867 + 1.19i)3-s + (−0.676 + 0.736i)4-s + (−1.14 − 0.372i)5-s + (1.44 + 0.314i)6-s + (−1.20 + 0.877i)7-s + (0.946 + 0.323i)8-s + (−0.363 − 1.11i)9-s + (0.119 + 1.19i)10-s + (0.965 − 0.262i)11-s + (−0.291 − 1.44i)12-s + (−0.115 + 0.0375i)13-s + (1.28 + 0.752i)14-s + (1.43 − 1.04i)15-s + (−0.0841 − 0.996i)16-s + (−0.179 + 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.495 - 0.868i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ -0.495 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124343 + 0.214080i\)
\(L(\frac12)\) \(\approx\) \(0.124343 + 0.214080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.568 + 1.29i)T \)
11 \( 1 + (-3.20 + 0.869i)T \)
good3 \( 1 + (1.50 - 2.06i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (2.56 + 0.832i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (3.19 - 2.32i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.416 - 0.135i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.739 - 2.27i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.14 - 4.32i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.35T + 23T^{2} \)
29 \( 1 + (-0.0866 - 0.119i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.489 - 1.50i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.33 - 5.97i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.92 - 2.12i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.62iT - 43T^{2} \)
47 \( 1 + (2.44 + 1.77i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.41 + 1.43i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.35 + 10.1i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.627 - 0.203i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 3.00iT - 67T^{2} \)
71 \( 1 + (3.26 - 10.0i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.61 - 6.25i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.99 + 6.13i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.17 + 0.705i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (-2.68 - 8.25i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.78631805625820012951863088984, −12.86242712329521448138952071114, −12.02667106942573535486732374789, −11.39797215634744999790071782391, −10.20316578162393755715305021263, −9.338724231683949912357241843356, −8.289643389352215305257501963210, −6.15294292476419634385509881037, −4.44537500692297947662959447656, −3.52332903246279044474576544399, 0.38431808398831852983422918476, 4.15729522642029389157562183486, 6.17628321064601706522686043665, 7.01630149578462089513554194792, 7.50058411883396804666083599113, 9.187286918422795870720474820137, 10.63969549726766931957020705650, 11.73765527323435889401129813051, 12.85049495444012124680149649148, 13.73861822639828209356018398717

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