Properties

Label 2-12e3-144.85-c1-0-6
Degree $2$
Conductor $1728$
Sign $0.675 - 0.737i$
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 + 0.267i)5-s + (−2.36 + 1.36i)7-s + (1.13 − 4.23i)11-s + (0.901 + 3.36i)13-s + 5.73·17-s + (2.36 − 2.36i)19-s + (−4.09 − 2.36i)23-s + (−3.40 + 1.96i)25-s + (−2.36 − 0.633i)29-s + (0.267 − 0.464i)31-s + (2 − 2i)35-s + (4.73 + 4.73i)37-s + (2.59 + 1.5i)41-s + (−2.23 + 8.33i)43-s + (3.83 + 6.63i)47-s + ⋯
L(s)  = 1  + (−0.447 + 0.119i)5-s + (−0.894 + 0.516i)7-s + (0.341 − 1.27i)11-s + (0.250 + 0.933i)13-s + 1.39·17-s + (0.542 − 0.542i)19-s + (−0.854 − 0.493i)23-s + (−0.680 + 0.392i)25-s + (−0.439 − 0.117i)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 + (−0.340 + 1.27i)43-s + (0.558 + 0.967i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.675 - 0.737i$
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: \((2,\ 1728,\ (\ :1/2),\ 0.675 - 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.323878026\)
\(L(\frac12)\) \(\approx\) \(1.323878026\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 0.267i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.13 + 4.23i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.901 - 3.36i)T + (-11.2 + 6.5i)T^{2} \)
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 \( 1 + (2.36 + 0.633i)T + (25.1 + 14.5i)T^{2} \)
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 \( 1 + (2.23 - 8.33i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.83 - 6.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.46 - 7.46i)T + 53iT^{2} \)
59 \( 1 + (-7.33 + 1.96i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-11.1 - 3i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.76 - 6.59i)T + (-58.0 + 33.5i)T^{2} \)
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 \( 1 + (1.36 + 0.366i)T + (71.8 + 41.5i)T^{2} \)
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

−9.513361868716215050264355851428, −8.624018377761520970286260698723, −7.929033643418317399410920539126, −7.00030876241595570568446644905, −6.08645823414107913493081756111, −5.65372280858847588804157806025, −4.25086069460542728494530302412, −3.46771461088538889382176181062, −2.65949281157556167479211193929, −1.01724971669487656938988415798, 0.63382429850806899140495433891, 2.10599638341692400929609899220, 3.62441142702721169725508933990, 3.82016279354230835189296415412, 5.23082665244508176365377287803, 5.92151720725035310854128803898, 7.02409312961649956867689365459, 7.57933830989433662417181241835, 8.257349537585372164621625702816, 9.449744643217422336775291920860

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