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Properties

Label 2-12e3-72.61-c1-0-17
Degree $2$
Conductor $1728$
Sign $0.669 + 0.742i$
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  + (1.5 − 0.866i)5-s + (1.80 − 3.12i)7-s + (0.635 + 0.367i)11-s + (0.527 − 0.304i)13-s + 5.52·17-s + 2i·19-s + (2.36 + 4.10i)23-s + (−1 + 1.73i)25-s + (6.78 + 3.91i)29-s + (−4.70 − 8.15i)31-s − 6.25i·35-s − 2.34i·37-s + (−4.26 − 7.38i)41-s + (−8.88 − 5.12i)43-s + (5.88 − 10.1i)47-s + ⋯
L(s)  = 1  + (0.670 − 0.387i)5-s + (0.682 − 1.18i)7-s + (0.191 + 0.110i)11-s + (0.146 − 0.0845i)13-s + 1.33·17-s + 0.458i·19-s + (0.493 + 0.855i)23-s + (−0.200 + 0.346i)25-s + (1.26 + 0.727i)29-s + (−0.845 − 1.46i)31-s − 1.05i·35-s − 0.384i·37-s + (−0.665 − 1.15i)41-s + (−1.35 − 0.782i)43-s + (0.857 − 1.48i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.669 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.255462653\)
\(L(\frac12)\) \(\approx\) \(2.255462653\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.80 + 3.12i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.635 - 0.367i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.527 + 0.304i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.52T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-2.36 - 4.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.78 - 3.91i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.70 + 8.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.34iT - 37T^{2} \)
41 \( 1 + (4.26 + 7.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.88 + 5.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.88 + 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.0iT - 53T^{2} \)
59 \( 1 + (-1.04 + 0.604i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.78 - 5.65i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.46 - 3.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.63T + 71T^{2} \)
73 \( 1 + 2.05T + 73T^{2} \)
79 \( 1 + (-1.24 + 2.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.6 + 6.12i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.94T + 89T^{2} \)
97 \( 1 + (-7.78 + 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.232442352953316467899714791828, −8.442260591130130454759194253414, −7.50452860262222383935338244750, −7.07031316757035801620767421117, −5.73787454367172235849370248830, −5.28345606145002491877191784896, −4.16606137136353017867088903233, −3.40007210077527992037391755648, −1.84294567568517132238363242505, −0.996009840429125663668194959930, 1.35828972256016386607465822268, 2.45955955057659299131689577230, 3.24553078595421609418624040794, 4.71633810620163766603590942938, 5.34639655684652868677337488489, 6.22435125121860150098228627359, 6.84310855746270092783017541306, 8.156619575192874972603473257972, 8.477131110453884732972347481501, 9.497684197080869367934346948781

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