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Properties

Label 2-12e3-144.13-c1-0-5
Degree $2$
Conductor $1728$
Sign $-0.0436 - 0.999i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.133 + 0.5i)5-s + (−2.13 + 1.23i)7-s + (−0.5 − 0.133i)11-s + (4.59 − 1.23i)13-s − 4·17-s + (3 + 3i)19-s + (−0.401 − 0.232i)23-s + (4.09 − 2.36i)25-s + (−0.866 + 3.23i)29-s + (−0.598 + 1.03i)31-s + (−0.901 − 0.901i)35-s + (−7.73 + 7.73i)37-s + (9.69 + 5.59i)41-s + (−8.69 − 2.33i)43-s + (4.59 + 7.96i)47-s + ⋯
L(s)  = 1  + (0.0599 + 0.223i)5-s + (−0.806 + 0.465i)7-s + (−0.150 − 0.0403i)11-s + (1.27 − 0.341i)13-s − 0.970·17-s + (0.688 + 0.688i)19-s + (−0.0838 − 0.0483i)23-s + (0.819 − 0.473i)25-s + (−0.160 + 0.600i)29-s + (−0.107 + 0.186i)31-s + (−0.152 − 0.152i)35-s + (−1.27 + 1.27i)37-s + (1.51 + 0.874i)41-s + (−1.32 − 0.355i)43-s + (0.670 + 1.16i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0436 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0436 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.0436 - 0.999i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.0436 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.253737652\)
\(L(\frac12)\) \(\approx\) \(1.253737652\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.133 - 0.5i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.13 - 1.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.133i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.59 + 1.23i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 + (0.401 + 0.232i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.866 - 3.23i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.598 - 1.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.73 - 7.73i)T - 37iT^{2} \)
41 \( 1 + (-9.69 - 5.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.69 + 2.33i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.59 - 7.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.26 - 2.26i)T - 53iT^{2} \)
59 \( 1 + (1.5 + 5.59i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.86 - 14.4i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.23 - 0.330i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 - 0.535iT - 73T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.16 + 11.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 11.8iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.436607754055324887100772631607, −8.767271757465129846167672293684, −8.079844636750965268019499013526, −6.98586089101479792969133037881, −6.29852304518973334001398976989, −5.65459380116066627054827743565, −4.55308548518387847710116764308, −3.43460322865860853170710630332, −2.77412209660101194973492465350, −1.32650019315594894267096988009, 0.49899007974051927870410641547, 1.93922166922713445916469938068, 3.25326565838557088583836862069, 3.98038449687516917327486690671, 5.00802930288598194060794768433, 5.97293396243320623592485619866, 6.76106074331466279356746353041, 7.37543836787487940866405481475, 8.525819093033071174761531904158, 9.096121856513327910326666135653

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