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Properties

Label 2-12e3-72.13-c1-0-14
Degree $2$
Conductor $1728$
Sign $0.742 + 0.669i$
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  + (1.5 + 0.866i)5-s + (−1.80 − 3.12i)7-s + (−0.635 + 0.367i)11-s + (0.527 + 0.304i)13-s + 5.52·17-s + 2i·19-s + (−2.36 + 4.10i)23-s + (−1 − 1.73i)25-s + (6.78 − 3.91i)29-s + (4.70 − 8.15i)31-s − 6.25i·35-s + 2.34i·37-s + (−4.26 + 7.38i)41-s + (8.88 − 5.12i)43-s + (−5.88 − 10.1i)47-s + ⋯
L(s)  = 1  + (0.670 + 0.387i)5-s + (−0.682 − 1.18i)7-s + (−0.191 + 0.110i)11-s + (0.146 + 0.0845i)13-s + 1.33·17-s + 0.458i·19-s + (−0.493 + 0.855i)23-s + (−0.200 − 0.346i)25-s + (1.26 − 0.727i)29-s + (0.845 − 1.46i)31-s − 1.05i·35-s + 0.384i·37-s + (−0.665 + 1.15i)41-s + (1.35 − 0.782i)43-s + (−0.857 − 1.48i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.742 + 0.669i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.742 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.780251220\)
\(L(\frac12)\) \(\approx\) \(1.780251220\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.80 + 3.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.635 - 0.367i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.527 - 0.304i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.52T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (2.36 - 4.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.78 + 3.91i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.70 + 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.34iT - 37T^{2} \)
41 \( 1 + (4.26 - 7.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.88 + 5.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.88 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.0iT - 53T^{2} \)
59 \( 1 + (1.04 + 0.604i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.78 + 5.65i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.46 - 3.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.63T + 71T^{2} \)
73 \( 1 + 2.05T + 73T^{2} \)
79 \( 1 + (1.24 + 2.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.6 + 6.12i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.94T + 89T^{2} \)
97 \( 1 + (-7.78 - 13.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.663497094875264747548407815067, −8.220740345241425318011206818594, −7.70944568402205313721131225921, −6.69079822233079387203942858348, −6.18053419985837898139528484242, −5.23035779857015164697686593018, −4.04185233228283877043812854259, −3.34311316659263361963122839582, −2.16466350155324272439538526679, −0.77373554605918909965692336427, 1.18738949407177946112074723341, 2.54521702014093961591703571450, 3.21239956619940656950488001428, 4.63005017359721214233872301665, 5.51639982626316865068709555613, 6.02207472514254415394214248562, 6.88050662230835292283888616138, 8.009956943283714105839920087633, 8.771124790012654148350667090547, 9.324300615039796420123087848662

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