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Properties

Label 2-88-11.4-c3-0-8
Degree $2$
Conductor $88$
Sign $-0.993 + 0.109i$
Analytic cond. $5.19216$
Root an. cond. $2.27863$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.02 − 6.21i)3-s + (−17.3 + 12.6i)5-s + (−9.35 − 28.7i)7-s + (−12.7 − 9.26i)9-s + (−27.5 + 23.9i)11-s + (−27.2 − 19.7i)13-s + (43.3 + 133. i)15-s + (−3.40 + 2.47i)17-s + (15.5 − 47.7i)19-s − 197.·21-s − 27.2·23-s + (103. − 318. i)25-s + (59.4 − 43.2i)27-s + (−7.51 − 23.1i)29-s + (−132. − 96.0i)31-s + ⋯
L(s)  = 1  + (0.388 − 1.19i)3-s + (−1.55 + 1.12i)5-s + (−0.504 − 1.55i)7-s + (−0.472 − 0.343i)9-s + (−0.755 + 0.655i)11-s + (−0.580 − 0.422i)13-s + (0.746 + 2.29i)15-s + (−0.0485 + 0.0352i)17-s + (0.187 − 0.576i)19-s − 2.05·21-s − 0.246·23-s + (0.828 − 2.54i)25-s + (0.423 − 0.307i)27-s + (−0.0481 − 0.148i)29-s + (−0.765 − 0.556i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $-0.993 + 0.109i$
Analytic conductor: \(5.19216\)
Root analytic conductor: \(2.27863\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :3/2),\ -0.993 + 0.109i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0306072 - 0.555824i\)
\(L(\frac12)\) \(\approx\) \(0.0306072 - 0.555824i\)
\(L(\frac{5}{2})\) 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 \( 1 + (27.5 - 23.9i)T \)
good3 \( 1 + (-2.02 + 6.21i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (17.3 - 12.6i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (9.35 + 28.7i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (27.2 + 19.7i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (3.40 - 2.47i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-15.5 + 47.7i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + 27.2T + 1.21e4T^{2} \)
29 \( 1 + (7.51 + 23.1i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (132. + 96.0i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (-96.0 - 295. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-124. + 381. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 - 216.T + 7.95e4T^{2} \)
47 \( 1 + (-31.5 + 96.9i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (190. + 138. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (39.0 + 120. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (206. - 150. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + 332.T + 3.00e5T^{2} \)
71 \( 1 + (273. - 198. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (-166. - 512. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-38.7 - 28.1i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-19.7 + 14.3i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + 1.49e3T + 7.04e5T^{2} \)
97 \( 1 + (-724. - 526. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−13.14921860885921560503736898468, −12.24726082179660229180611590371, −11.00758719698865483595372744611, −10.13611919997651020238545049243, −7.939370221290429074520758203845, −7.38531887833777044535158336603, −6.83801611618513231828769464410, −4.18251840142318773081489374316, −2.81202363370593926608147125136, −0.30896169002472842178713129213, 3.15656343576287382206357898660, 4.42826013215091722393491553447, 5.52748367696184595338870571464, 7.81973054686149543531310118947, 8.836439510366847845561638250795, 9.423253912365042449869323270587, 11.02389629483686473429552488659, 12.13633650336726353284790799191, 12.76559837960113954309572929550, 14.63500254627137093636591707977

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