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Properties

Label 2-12e3-9.2-c2-0-11
Degree $2$
Conductor $1728$
Sign $0.743 - 0.668i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.20 + 4.73i)5-s + (−1.05 + 1.83i)7-s + (−13.7 − 7.91i)11-s + (−4.70 − 8.14i)13-s − 11.6i·17-s + 12.9·19-s + (−5.27 + 3.04i)23-s + (32.4 − 56.1i)25-s + (−24.7 − 14.2i)29-s + (−8.75 − 15.1i)31-s − 20.0i·35-s + 15.6·37-s + (−14.8 + 8.54i)41-s + (−21.7 + 37.6i)43-s + (20.6 + 11.9i)47-s + ⋯
L(s)  = 1  + (−1.64 + 0.947i)5-s + (−0.150 + 0.261i)7-s + (−1.24 − 0.719i)11-s + (−0.361 − 0.626i)13-s − 0.682i·17-s + 0.682·19-s + (−0.229 + 0.132i)23-s + (1.29 − 2.24i)25-s + (−0.854 − 0.493i)29-s + (−0.282 − 0.489i)31-s − 0.572i·35-s + 0.422·37-s + (−0.361 + 0.208i)41-s + (−0.505 + 0.874i)43-s + (0.439 + 0.253i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.743 - 0.668i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.743 - 0.668i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6747986023\)
\(L(\frac12)\) \(\approx\) \(0.6747986023\)
\(L(2)\) 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 \( 1 + (8.20 - 4.73i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (1.05 - 1.83i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (13.7 + 7.91i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (4.70 + 8.14i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 11.6iT - 289T^{2} \)
19 \( 1 - 12.9T + 361T^{2} \)
23 \( 1 + (5.27 - 3.04i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (24.7 + 14.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (8.75 + 15.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 15.6T + 1.36e3T^{2} \)
41 \( 1 + (14.8 - 8.54i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (21.7 - 37.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-20.6 - 11.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 14.1iT - 2.80e3T^{2} \)
59 \( 1 + (38.5 - 22.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-1.86 + 3.22i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-21.0 - 36.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 120. iT - 5.04e3T^{2} \)
73 \( 1 - 5.48T + 5.32e3T^{2} \)
79 \( 1 + (60.5 - 104. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (46.5 + 26.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 102. iT - 7.92e3T^{2} \)
97 \( 1 + (58.9 - 102. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.202617537290130789105562310572, −7.979176069754582720700992087292, −7.84237157258092729240241083445, −7.08855829186403118161997799637, −6.01286622652858296054666208026, −5.13342294744799433621844475023, −4.08679686265089475595422622545, −3.11749171821479223255776252988, −2.67305193790346076097972474633, −0.51173296511356399201294720288, 0.37176663470732499498831274930, 1.80445464429976423030723350698, 3.26966803884910400060778369421, 4.09224633328407804940748018355, 4.82200555559853935844080472071, 5.53165616241085943391981880230, 7.09390918329374495840907208396, 7.41065983461506264701750974979, 8.252301497085226689299406007377, 8.835423060851422742764339167746

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