Properties

Label 2-12e3-8.3-c0-0-5
Degree $2$
Conductor $1728$
Sign $0.707 + 0.707i$
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  − 1.73i·5-s + i·7-s + 1.73·11-s − 1.99·25-s i·31-s + 1.73·35-s − 1.73i·53-s − 2.99i·55-s + 73-s + 1.73i·77-s + 2i·79-s − 1.73·83-s − 97-s − 1.73i·101-s + 2i·103-s + ⋯
L(s)  = 1  − 1.73i·5-s + i·7-s + 1.73·11-s − 1.99·25-s i·31-s + 1.73·35-s − 1.73i·53-s − 2.99i·55-s + 73-s + 1.73i·77-s + 2i·79-s − 1.73·83-s − 97-s − 1.73i·101-s + 2i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.707 + 0.707i$
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: \((2,\ 1728,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226193708\)
\(L(\frac12)\) \(\approx\) \(1.226193708\)
\(L(1)\) 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 \( 1 + 1.73iT - T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.73iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
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

−9.460297355588481295784478368472, −8.586311973124528958696126120511, −8.259837785756685271928415424213, −6.96554693003475030939259925085, −6.01058269903611076093421220034, −5.36461389378710630803584534790, −4.46875602335939669220137224221, −3.72500822319344091911680340217, −2.15091601406651104804112614109, −1.12668990585797744293638940924, 1.48913112638116237446730664632, 2.87473423681427240939004859718, 3.69020245367746803452221434075, 4.35252477490520141037581438516, 5.85216650966682629769221314303, 6.69667009869626762415287951612, 6.99490072783038497509693286017, 7.79244623976972059789088440157, 8.936986170836669074269077289324, 9.736917408351408541347054304305

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