Properties

Label 2-864-12.11-c1-0-15
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  − 3.86i·5-s − 3.73i·7-s − 1.03·11-s + 4.46·13-s + 1.79i·17-s + 1.73i·19-s − 8.76·23-s − 9.92·25-s − 7.72i·29-s + 7.46i·31-s − 14.4·35-s + 0.464·37-s − 7.72i·41-s + 0.535i·43-s + 4.62·47-s + ⋯
L(s)  = 1  − 1.72i·5-s − 1.41i·7-s − 0.312·11-s + 1.23·13-s + 0.434i·17-s + 0.397i·19-s − 1.82·23-s − 1.98·25-s − 1.43i·29-s + 1.34i·31-s − 2.43·35-s + 0.0762·37-s − 1.20i·41-s + 0.0817i·43-s + 0.674·47-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

\(L(1)\) \(\approx\) \(0.507881 - 1.22613i\)
\(L(\frac12)\) \(\approx\) \(0.507881 - 1.22613i\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.86iT - 5T^{2} \)
7 \( 1 + 3.73iT - 7T^{2} \)
11 \( 1 + 1.03T + 11T^{2} \)
13 \( 1 - 4.46T + 13T^{2} \)
17 \( 1 - 1.79iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + 8.76T + 23T^{2} \)
29 \( 1 + 7.72iT - 29T^{2} \)
31 \( 1 - 7.46iT - 31T^{2} \)
37 \( 1 - 0.464T + 37T^{2} \)
41 \( 1 + 7.72iT - 41T^{2} \)
43 \( 1 - 0.535iT - 43T^{2} \)
47 \( 1 - 4.62T + 47T^{2} \)
53 \( 1 - 3.58iT - 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 6.26iT - 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 3.92T + 73T^{2} \)
79 \( 1 + 4.80iT - 79T^{2} \)
83 \( 1 + 2.07T + 83T^{2} \)
89 \( 1 - 1.79iT - 89T^{2} \)
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

−9.974629159830348264352272113169, −8.823523064934342018288973399602, −8.251332183775759474952069077601, −7.52431557931234262523086836534, −6.23042011294747290996461380007, −5.41152345835557600566557292680, −4.20459213742141997716349652252, −3.87214533886772105797350520828, −1.72140704728848553966657868907, −0.64975664833088037213141356954, 2.14799318550522853569838822839, 2.92083672872112729719295680530, 3.92860801786674074445109687465, 5.54069859804751479321771643303, 6.13171556412645686313583677823, 6.93552088190976191856954141683, 7.957133310168008246474097661904, 8.756640053941827743486306984747, 9.773965477762513294556883293246, 10.47707732348526822397708799774

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