Properties

Label 2-12e3-144.85-c1-0-17
Degree $2$
Conductor $1728$
Sign $-0.216 + 0.976i$
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 + 0.133i)5-s + (2.13 − 1.23i)7-s + (0.133 − 0.5i)11-s + (−1.23 − 4.59i)13-s − 4·17-s + (3 − 3i)19-s + (0.401 + 0.232i)23-s + (−4.09 + 2.36i)25-s + (3.23 + 0.866i)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 + (2.33 − 8.69i)43-s + (4.59 + 7.96i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.0599i)5-s + (0.806 − 0.465i)7-s + (0.0403 − 0.150i)11-s + (−0.341 − 1.27i)13-s − 0.970·17-s + (0.688 − 0.688i)19-s + (0.0838 + 0.0483i)23-s + (−0.819 + 0.473i)25-s + (0.600 + 0.160i)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 + (0.355 − 1.32i)43-s + (0.670 + 1.16i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.216 + 0.976i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.216 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.297964728\)
\(L(\frac12)\) \(\approx\) \(1.297964728\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.133i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.13 + 1.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.133 + 0.5i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.23 + 4.59i)T + (-11.2 + 6.5i)T^{2} \)
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 \( 1 + (-3.23 - 0.866i)T + (25.1 + 14.5i)T^{2} \)
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 \( 1 + (-2.33 + 8.69i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.59 - 7.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.26 + 2.26i)T + 53iT^{2} \)
59 \( 1 + (-5.59 + 1.5i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-14.4 - 3.86i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.330 - 1.23i)T + (-58.0 + 33.5i)T^{2} \)
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 \( 1 + (11.7 + 3.16i)T + (71.8 + 41.5i)T^{2} \)
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

−8.942012123957885371245305092249, −8.333426175876869398989167860180, −7.41045820715237339627714819080, −6.99624551693082218763962175381, −5.65066705617742639410644047192, −5.06295189108753553728248206334, −4.07274578880151473524991261477, −3.12247372255735707840465447485, −1.92178264714358623691457130842, −0.48408537918084996282703278621, 1.54016217719993292087881043581, 2.43156374927458278387823208067, 3.79488360529797966489264682391, 4.63541389630485940633144160847, 5.33450123377422134380977000547, 6.48205932740724701371288312357, 7.08190919634354217531699761510, 8.217574445028308497749107375548, 8.538365424307790909145121557904, 9.592405203323529378797681742954

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