Properties

Label 2-12e3-8.3-c2-0-63
Degree $2$
Conductor $1728$
Sign $-0.707 - 0.707i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.63i·5-s − 13.4i·7-s − 18.4·11-s − 18.9·25-s − 29.3i·29-s − 61.4i·31-s − 89.4·35-s − 132.·49-s + 105. i·53-s + 122. i·55-s + 117.·59-s + 143.·73-s + 248. i·77-s − 58i·79-s − 165.·83-s + ⋯
L(s)  = 1  − 1.32i·5-s − 1.92i·7-s − 1.67·11-s − 0.758·25-s − 1.01i·29-s − 1.98i·31-s − 2.55·35-s − 2.71·49-s + 1.99i·53-s + 2.22i·55-s + 1.99·59-s + 1.96·73-s + 3.23i·77-s − 0.734i·79-s − 1.98·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9200338645\)
\(L(\frac12)\) \(\approx\) \(0.9200338645\)
\(L(2)\) 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 \( 1 + 6.63iT - 25T^{2} \)
7 \( 1 + 13.4iT - 49T^{2} \)
11 \( 1 + 18.4T + 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + 29.3iT - 841T^{2} \)
31 \( 1 + 61.4iT - 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 - 105. iT - 2.80e3T^{2} \)
59 \( 1 - 117.T + 3.48e3T^{2} \)
61 \( 1 - 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 143.T + 5.32e3T^{2} \)
79 \( 1 + 58iT - 6.24e3T^{2} \)
83 \( 1 + 165.T + 6.88e3T^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 - 128.T + 9.40e3T^{2} \)
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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

−8.475969012776861279342415559234, −7.77174818194062451736951000899, −7.39793098302283196904375365547, −6.13880432051641580775514142678, −5.18694342241594614285157440266, −4.47171985579669143614045262240, −3.81763647901023246962025327102, −2.39550814104313377710871444034, −0.987190930114747468350951789690, −0.27469055374492088567566089264, 2.04010529517000824752362310769, 2.75684779485349219653072775597, 3.33908590069716866549716745599, 5.12876994734724786786549407145, 5.42796900465872366107340795070, 6.49226646980777459113875287998, 7.12835143813770675967687164779, 8.218601955460601789227695407983, 8.661018832761980983147355736189, 9.731197487201931588337897663737

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