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Properties

Label 2-864-108.11-c1-0-26
Degree $2$
Conductor $864$
Sign $0.991 - 0.129i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
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  + (1.23 + 1.21i)3-s + (0.898 − 2.46i)5-s + (2.70 + 0.477i)7-s + (0.0269 + 2.99i)9-s + (3.89 − 1.41i)11-s + (−0.124 + 0.104i)13-s + (4.11 − 1.94i)15-s + (−4.20 + 2.43i)17-s + (−2.41 − 1.39i)19-s + (2.75 + 3.89i)21-s + (−0.763 − 4.33i)23-s + (−1.45 − 1.21i)25-s + (−3.62 + 3.72i)27-s + (4.22 − 5.03i)29-s + (1.23 − 0.216i)31-s + ⋯
L(s)  = 1  + (0.710 + 0.703i)3-s + (0.401 − 1.10i)5-s + (1.02 + 0.180i)7-s + (0.00898 + 0.999i)9-s + (1.17 − 0.427i)11-s + (−0.0345 + 0.0290i)13-s + (1.06 − 0.501i)15-s + (−1.02 + 0.589i)17-s + (−0.555 − 0.320i)19-s + (0.600 + 0.849i)21-s + (−0.159 − 0.903i)23-s + (−0.290 − 0.243i)25-s + (−0.697 + 0.716i)27-s + (0.784 − 0.934i)29-s + (0.221 − 0.0389i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.991 - 0.129i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42005 + 0.157325i\)
\(L(\frac12)\) \(\approx\) \(2.42005 + 0.157325i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.23 - 1.21i)T \)
good5 \( 1 + (-0.898 + 2.46i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-2.70 - 0.477i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-3.89 + 1.41i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.124 - 0.104i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (4.20 - 2.43i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.41 + 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.763 + 4.33i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-4.22 + 5.03i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.23 + 0.216i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.72 - 4.72i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.17 + 3.78i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.20 - 6.05i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.18 + 6.72i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + (-7.26 - 2.64i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.610 - 3.46i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (9.98 + 11.8i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-5.22 - 9.05i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.26 - 12.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.58 - 4.27i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-5.75 - 4.83i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (7.95 + 4.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.0 - 5.85i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.03577106415681645028248419828, −9.102374291318807783064684547077, −8.586079236415521942479671532514, −8.165828564058591055368604440550, −6.68601848036498247356386293483, −5.61108789346434549975847293872, −4.49235641497111242255943583615, −4.23666544799182028916681886652, −2.53094264286986422555482510144, −1.40682022020945296640451746188, 1.51964165892691402667386009463, 2.40788563369074394222161098942, 3.58000102848965273074434760968, 4.66080280915736641912285882218, 6.14368665824745253428458582194, 6.85098028797528759648597316559, 7.43759575175257289487651562169, 8.454714962408013235191500096167, 9.199795650076020612116980289837, 10.09367009534846710498901014080

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