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Properties

Label 2-864-12.11-c1-0-6
Degree $2$
Conductor $864$
Sign $0.707 - 0.707i$
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  + 1.03i·5-s − 0.267i·7-s + 3.86·11-s − 2.46·13-s + 6.69i·17-s − 1.73i·19-s + 5.93·23-s + 3.92·25-s + 2.07i·29-s + 0.535i·31-s + 0.277·35-s − 6.46·37-s + 2.07i·41-s + 7.46i·43-s + 9.52·47-s + ⋯
L(s)  = 1  + 0.462i·5-s − 0.101i·7-s + 1.16·11-s − 0.683·13-s + 1.62i·17-s − 0.397i·19-s + 1.23·23-s + 0.785·25-s + 0.384i·29-s + 0.0962i·31-s + 0.0468·35-s − 1.06·37-s + 0.323i·41-s + 1.13i·43-s + 1.38·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47060 + 0.609143i\)
\(L(\frac12)\) \(\approx\) \(1.47060 + 0.609143i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 1.03iT - 5T^{2} \)
7 \( 1 + 0.267iT - 7T^{2} \)
11 \( 1 - 3.86T + 11T^{2} \)
13 \( 1 + 2.46T + 13T^{2} \)
17 \( 1 - 6.69iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 - 5.93T + 23T^{2} \)
29 \( 1 - 2.07iT - 29T^{2} \)
31 \( 1 - 0.535iT - 31T^{2} \)
37 \( 1 + 6.46T + 37T^{2} \)
41 \( 1 - 2.07iT - 41T^{2} \)
43 \( 1 - 7.46iT - 43T^{2} \)
47 \( 1 - 9.52T + 47T^{2} \)
53 \( 1 - 13.3iT - 53T^{2} \)
59 \( 1 - 7.45T + 59T^{2} \)
61 \( 1 + 9.39T + 61T^{2} \)
67 \( 1 + 9.73iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 9.92T + 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 - 7.72T + 83T^{2} \)
89 \( 1 - 6.69iT - 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−10.52789043504298683632042710640, −9.271550924102430695834167721685, −8.787859761680774186759579141853, −7.58949198898553359724077740503, −6.81927772251264904429896955373, −6.09155741461787290003463123102, −4.87247457617568423749775393389, −3.88496672397657344866437606127, −2.82902844697817755613214406137, −1.37798657253915116309520554082, 0.894144842261612430244627741133, 2.43349781213787736834270861955, 3.68311921370787232084537428104, 4.79815140792626942212772814233, 5.50120199367764552386343385053, 6.84106904349065576221345472674, 7.28604765276901643802447665460, 8.639530640939996404257894471203, 9.130264051920336664175626982192, 9.914834395762176510665904349323

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