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Properties

Label 2-12e3-9.7-c1-0-18
Degree $2$
Conductor $1728$
Sign $-0.642 + 0.766i$
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  + (0.5 + 0.866i)5-s + (0.866 − 1.5i)7-s + (−0.866 + 1.5i)11-s + (−1.5 − 2.59i)13-s − 4·17-s − 6.92·19-s + (−4.33 − 7.5i)23-s + (2 − 3.46i)25-s + (−0.5 + 0.866i)29-s + (−2.59 − 4.5i)31-s + 1.73·35-s + 8·37-s + (2.5 + 4.33i)41-s + (−4.33 + 7.5i)43-s + (6.06 − 10.5i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.327 − 0.566i)7-s + (−0.261 + 0.452i)11-s + (−0.416 − 0.720i)13-s − 0.970·17-s − 1.58·19-s + (−0.902 − 1.56i)23-s + (0.400 − 0.692i)25-s + (−0.0928 + 0.160i)29-s + (−0.466 − 0.808i)31-s + 0.292·35-s + 1.31·37-s + (0.390 + 0.676i)41-s + (−0.660 + 1.14i)43-s + (0.884 − 1.53i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7568517982\)
\(L(\frac12)\) \(\approx\) \(0.7568517982\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.866 + 1.5i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.866 - 1.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 + (4.33 + 7.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.59 + 4.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.33 - 7.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.06 + 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 + 7.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + (-2.59 + 4.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.33 + 7.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + (-1.5 + 2.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.981380771192917287850770762128, −8.171056583087425381518915097330, −7.52315503629799124877371750438, −6.52970715082192336452072885772, −6.03406668867241788693632621365, −4.58178110302276955048212056772, −4.32239428810879200709165230733, −2.78268414960455543237508180676, −2.03880472520197407921652953260, −0.26139803985808761942234670397, 1.65558859901771308675742444089, 2.48719024895437509774433618824, 3.84851659548527075717482829865, 4.69760045740092967387864654559, 5.57482034236152313071981363797, 6.29508088475627907043480150340, 7.26216866899246222282810857652, 8.129058634302893145455878479030, 8.988918065143951584200626676733, 9.283814708581792040269399254961

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