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Properties

Label 2-12e3-8.3-c0-0-4
Degree $2$
Conductor $1728$
Sign $0.965 + 0.258i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·7-s + 1.73i·13-s + 1.73·19-s + 25-s − 2i·31-s − 1.73i·37-s + 1.73i·61-s − 1.73·67-s + 73-s + i·79-s + 1.73·91-s − 97-s + i·103-s + ⋯
L(s)  = 1  − i·7-s + 1.73i·13-s + 1.73·19-s + 25-s − 2i·31-s − 1.73i·37-s + 1.73i·61-s − 1.73·67-s + 73-s + i·79-s + 1.73·91-s − 97-s + i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.965 + 0.258i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.965 + 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179028814\)
\(L(\frac12)\) \(\approx\) \(1.179028814\)
\(L(1)\) 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 \( 1 - T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.73T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.73iT - T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.421939155349176330696091208344, −8.894557769454848325364965885379, −7.60362536570571617019310146860, −7.26970994478157297876360584353, −6.38398587584387894507570288870, −5.38873634716442405665106788222, −4.34475742944151986063552940965, −3.77713839038321684515577177177, −2.46821184475163916363422725550, −1.15771038397960263291517791310, 1.28671355777553037341701740716, 2.89939982696923096856465047296, 3.24892272853930578038120055939, 4.97030408834250382418989449407, 5.31772185895697473213617645467, 6.26291753977632947805453132930, 7.22375503157294856554603009801, 8.080881452888941504927883628327, 8.687828743886921354485427057203, 9.532573045700781817757785491729

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