Properties

Label 2-12e3-36.7-c0-0-0
Degree $2$
Conductor $1728$
Sign $0.642 - 0.766i$
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  + (0.5 + 0.866i)5-s + (−0.866 − 0.5i)7-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s − 0.999i·35-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 + 0.5i)47-s + 0.999i·55-s + (−0.866 + 0.5i)59-s + (0.5 − 0.866i)61-s + (−0.499 + 0.866i)65-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (−0.866 − 0.5i)7-s + (0.866 + 0.5i)11-s + (0.5 + 0.866i)13-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)29-s + (0.866 − 0.5i)31-s − 0.999i·35-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 + 0.5i)47-s + 0.999i·55-s + (−0.866 + 0.5i)59-s + (0.5 − 0.866i)61-s + (−0.499 + 0.866i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167352422\)
\(L(\frac12)\) \(\approx\) \(1.167352422\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.543885514652041122002703582331, −9.175082324694697094749423906716, −7.914398135782110932434010822138, −7.03004020555584160085217191519, −6.45671470629239178836423714981, −5.93230029871973798035470081817, −4.46405782510757234638656158648, −3.73866777472437973757336750750, −2.74950149137168911424797330324, −1.56310958133187847681320844723, 0.987392758379767511786723518841, 2.38665520872311839944329547963, 3.48679898519357348431707504018, 4.38707309781617242343030375863, 5.68389307123091056654112544851, 5.90081304946985317127610828765, 6.87934197690069529202837490502, 8.034300385598556594363479692767, 8.777892950440157883771614432740, 9.271426803935161031373606424733

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