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Properties

Label 2-12e3-144.11-c1-0-21
Degree $2$
Conductor $1728$
Sign $-0.874 - 0.484i$
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  + (−3.73 − 1.00i)5-s + (1.68 − 2.91i)7-s + (0.211 − 0.0566i)11-s + (−2.71 − 0.727i)13-s − 4.23i·17-s + (1.12 + 1.12i)19-s + (3.33 − 1.92i)23-s + (8.61 + 4.97i)25-s + (−2.03 + 0.545i)29-s + (−7.21 + 4.16i)31-s + (−9.19 + 9.19i)35-s + (−2.66 − 2.66i)37-s + (−1.70 − 2.95i)41-s + (1.25 + 4.68i)43-s + (−2.34 + 4.07i)47-s + ⋯
L(s)  = 1  + (−1.67 − 0.447i)5-s + (0.635 − 1.10i)7-s + (0.0637 − 0.0170i)11-s + (−0.753 − 0.201i)13-s − 1.02i·17-s + (0.257 + 0.257i)19-s + (0.695 − 0.401i)23-s + (1.72 + 0.994i)25-s + (−0.378 + 0.101i)29-s + (−1.29 + 0.747i)31-s + (−1.55 + 1.55i)35-s + (−0.438 − 0.438i)37-s + (−0.266 − 0.460i)41-s + (0.191 + 0.715i)43-s + (−0.342 + 0.593i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.874 - 0.484i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.874 - 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2115269502\)
\(L(\frac12)\) \(\approx\) \(0.2115269502\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.73 + 1.00i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.68 + 2.91i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.211 + 0.0566i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.71 + 0.727i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 4.23iT - 17T^{2} \)
19 \( 1 + (-1.12 - 1.12i)T + 19iT^{2} \)
23 \( 1 + (-3.33 + 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.03 - 0.545i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (7.21 - 4.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.66 + 2.66i)T + 37iT^{2} \)
41 \( 1 + (1.70 + 2.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.25 - 4.68i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.34 - 4.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.58 - 7.58i)T - 53iT^{2} \)
59 \( 1 + (1.43 - 5.34i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.33 + 8.69i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.38 - 5.17i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.53iT - 71T^{2} \)
73 \( 1 - 3.22iT - 73T^{2} \)
79 \( 1 + (4.98 + 2.87i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.20 - 4.50i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.96T + 89T^{2} \)
97 \( 1 + (7.63 - 13.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.782960427154178726088167223472, −7.76966352240738497600034941270, −7.52627638698231522449911919149, −6.84196591256098770581975948535, −5.23636363060565239677191996252, −4.65094794987246623760404220910, −3.90516496612573601121153323211, −3.01471136469770027950498492317, −1.24399758858622265223281049470, −0.087094366080996813251654995425, 1.86612129566190619320805714542, 3.04935972382026400115966411619, 3.89451237179498765553591173857, 4.81387014988192831888561168280, 5.62976799267629009661167997952, 6.81117385876008959567569733313, 7.48389009241330843687500357774, 8.189230524891376381574066336782, 8.776596285767031297679432551844, 9.680779477394810271261843487714

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