Properties

Label 2-88-88.69-c1-0-6
Degree $2$
Conductor $88$
Sign $0.802 + 0.596i$
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  + (−1.26 − 0.632i)2-s + (2.32 − 0.756i)3-s + (1.20 + 1.59i)4-s + (0.117 − 0.162i)5-s + (−3.42 − 0.515i)6-s + (−0.725 + 2.23i)7-s + (−0.507 − 2.78i)8-s + (2.42 − 1.75i)9-s + (−0.251 + 0.130i)10-s + (−1.47 − 2.97i)11-s + (4.00 + 2.81i)12-s + (−0.959 − 1.31i)13-s + (2.32 − 2.36i)14-s + (0.151 − 0.466i)15-s + (−1.11 + 3.84i)16-s + (2.69 + 1.96i)17-s + ⋯
L(s)  = 1  + (−0.894 − 0.447i)2-s + (1.34 − 0.436i)3-s + (0.600 + 0.799i)4-s + (0.0526 − 0.0725i)5-s + (−1.39 − 0.210i)6-s + (−0.274 + 0.843i)7-s + (−0.179 − 0.983i)8-s + (0.806 − 0.586i)9-s + (−0.0795 + 0.0413i)10-s + (−0.445 − 0.895i)11-s + (1.15 + 0.812i)12-s + (−0.265 − 0.366i)13-s + (0.622 − 0.631i)14-s + (0.0391 − 0.120i)15-s + (−0.279 + 0.960i)16-s + (0.654 + 0.475i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88\)    =    \(2^{3} \cdot 11\)
Sign: $0.802 + 0.596i$
Analytic conductor: \(0.702683\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{88} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 88,\ (\ :1/2),\ 0.802 + 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884919 - 0.292740i\)
\(L(\frac12)\) \(\approx\) \(0.884919 - 0.292740i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.632i)T \)
11 \( 1 + (1.47 + 2.97i)T \)
good3 \( 1 + (-2.32 + 0.756i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.117 + 0.162i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.725 - 2.23i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.959 + 1.31i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.69 - 1.96i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.68 - 1.52i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 9.25T + 23T^{2} \)
29 \( 1 + (-5.88 - 1.91i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.68 + 4.85i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.21 + 0.395i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.86 + 5.73i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.47iT - 43T^{2} \)
47 \( 1 + (-1.47 - 4.52i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.14 + 7.08i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.81 + 0.588i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.75 + 5.17i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 8.27iT - 67T^{2} \)
71 \( 1 + (-1.00 - 0.733i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.13 + 6.55i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.8 + 8.59i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.50 - 7.58i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 5.11T + 89T^{2} \)
97 \( 1 + (0.242 - 0.176i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.97221356888356010391053643182, −12.87227442286591973564974052326, −12.06761763041847753915235776243, −10.53671925223879893159656400696, −9.441987834283141943617041493682, −8.359667480506134675809153734659, −7.941984579725435262317243040986, −6.18429973400840359177159105907, −3.40216606124835659086369817134, −2.22284984825841306938549309746, 2.48076926623779336449734768726, 4.42982898716618090200949928821, 6.60734522678314449446653658948, 7.78080408566493301386570682025, 8.623221255630813577927732873989, 10.06851530593758034089948303887, 10.13065867878245770621573886778, 12.08699573433454703334046173609, 13.79240734502269986166861011196, 14.34581603489732820815647135488

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