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Properties

Label 2-12e3-24.5-c0-0-0
Degree $2$
Conductor $1728$
Sign $-0.258 - 0.965i$
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  − 1.73·7-s + 1.73i·13-s + i·19-s − 25-s + 1.73i·37-s + 2i·43-s + 1.99·49-s − 1.73i·61-s − i·67-s − 73-s − 1.73·79-s − 2.99i·91-s + 97-s + 1.73·103-s + ⋯
L(s)  = 1  − 1.73·7-s + 1.73i·13-s + i·19-s − 25-s + 1.73i·37-s + 2i·43-s + 1.99·49-s − 1.73i·61-s − i·67-s − 73-s − 1.73·79-s − 2.99i·91-s + 97-s + 1.73·103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ -0.258 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6342435464\)
\(L(\frac12)\) \(\approx\) \(0.6342435464\)
\(L(1)\) 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 \( 1 + 1.73T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.73iT - T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + 1.73T + T^{2} \)
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

−9.702820901125660573436345466720, −9.183040230168732113106534547477, −8.203069295365769715082556465148, −7.23398982359682651920682043386, −6.36512830745111480021899452796, −6.10369930287206406779904534040, −4.69007782614118770680168609932, −3.80692219246757224978804995502, −2.99981303877586771403430792425, −1.72280138499637881989110314377, 0.46338051862625487952607940924, 2.48014192037986862179197496554, 3.25591163728466483437690393791, 4.08379331213660696936032823688, 5.51353532999741307869215834485, 5.90508806644719562779067785606, 6.99326595092749575480987268386, 7.51821779879046723797360365830, 8.669301247567022146980044254525, 9.265227174105815900397456006040

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