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Properties

Label 2-12e3-144.133-c1-0-9
Degree $2$
Conductor $1728$
Sign $0.999 + 0.0436i$
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  + (0.267 − i)5-s + (2.36 + 1.36i)7-s + (−4.23 + 1.13i)11-s + (−3.36 − 0.901i)13-s + 5.73·17-s + (2.36 − 2.36i)19-s + (4.09 − 2.36i)23-s + (3.40 + 1.96i)25-s + (0.633 + 2.36i)29-s + (0.267 + 0.464i)31-s + (2 − 2i)35-s + (4.73 + 4.73i)37-s + (−2.59 + 1.5i)41-s + (8.33 − 2.23i)43-s + (3.83 − 6.63i)47-s + ⋯
L(s)  = 1  + (0.119 − 0.447i)5-s + (0.894 + 0.516i)7-s + (−1.27 + 0.341i)11-s + (−0.933 − 0.250i)13-s + 1.39·17-s + (0.542 − 0.542i)19-s + (0.854 − 0.493i)23-s + (0.680 + 0.392i)25-s + (0.117 + 0.439i)29-s + (0.0481 + 0.0833i)31-s + (0.338 − 0.338i)35-s + (0.777 + 0.777i)37-s + (−0.405 + 0.234i)41-s + (1.27 − 0.340i)43-s + (0.558 − 0.967i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.999 + 0.0436i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1585, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.999 + 0.0436i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.872246260\)
\(L(\frac12)\) \(\approx\) \(1.872246260\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.267 + i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.36 - 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.23 - 1.13i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.36 + 0.901i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 5.73T + 17T^{2} \)
19 \( 1 + (-2.36 + 2.36i)T - 19iT^{2} \)
23 \( 1 + (-4.09 + 2.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.633 - 2.36i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.267 - 0.464i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.73 - 4.73i)T + 37iT^{2} \)
41 \( 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.33 + 2.23i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.83 + 6.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.46 - 7.46i)T + 53iT^{2} \)
59 \( 1 + (1.96 - 7.33i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3 + 11.1i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (6.59 + 1.76i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 - 6.26iT - 73T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.366 - 1.36i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2iT - 89T^{2} \)
97 \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.234775883972032764507275598390, −8.530430753452942415992497557173, −7.66745783278781182941046060130, −7.24957082360620545066372663309, −5.82278684955675445753518697344, −5.01393902384560110818772415562, −4.82157672420093057062033207925, −3.13671282698050593334454259811, −2.34044043373154937088283611177, −0.981273234775047120672720216413, 0.972848003299635026864511735236, 2.39243781198595182523525984267, 3.23911703973981658154019301280, 4.45945386841373724524389635188, 5.26038041375539002259844229219, 5.94502945978198527284105974404, 7.36192997897422446276226088744, 7.52761367952931196035172671127, 8.352712515594923620675184241259, 9.468392550522869192553031955239

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