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Properties

Label 2-12e3-216.101-c0-0-1
Degree $2$
Conductor $1728$
Sign $0.524 + 0.851i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 − 0.173i)3-s + (0.939 + 0.342i)9-s + (0.223 − 1.26i)11-s + (−0.592 + 0.342i)17-s + (−0.300 − 0.173i)19-s + (0.939 − 0.342i)25-s + (−0.866 − 0.5i)27-s + (−0.439 + 1.20i)33-s + (0.673 − 1.85i)41-s + (1.85 + 0.326i)43-s + (−0.173 − 0.984i)49-s + (0.642 − 0.233i)51-s + (0.266 + 0.223i)57-s + (−0.342 − 1.93i)59-s + (−0.524 + 1.43i)67-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (0.939 + 0.342i)9-s + (0.223 − 1.26i)11-s + (−0.592 + 0.342i)17-s + (−0.300 − 0.173i)19-s + (0.939 − 0.342i)25-s + (−0.866 − 0.5i)27-s + (−0.439 + 1.20i)33-s + (0.673 − 1.85i)41-s + (1.85 + 0.326i)43-s + (−0.173 − 0.984i)49-s + (0.642 − 0.233i)51-s + (0.266 + 0.223i)57-s + (−0.342 − 1.93i)59-s + (−0.524 + 1.43i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.524 + 0.851i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.524 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7807790079\)
\(L(\frac12)\) \(\approx\) \(0.7807790079\)
\(L(1)\) 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 \( 1 + (0.984 + 0.173i)T \)
good5 \( 1 + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.173 + 0.984i)T^{2} \)
11 \( 1 + (-0.223 + 1.26i)T + (-0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.300 + 0.173i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.173 - 0.984i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.673 + 1.85i)T + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (-1.85 - 0.326i)T + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.342 + 1.93i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (-1.62 + 0.592i)T + (0.766 - 0.642i)T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
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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

−9.320610515133337308406945965383, −8.646998266931391904395524963436, −7.72977894069639691896833997715, −6.79628325393992744385928196894, −6.17396500609062790985967130027, −5.44817133417349043195611961484, −4.51194612897286816171435651809, −3.58610971307695074861548511999, −2.22286064974075483160815872119, −0.76384933118846871227632830053, 1.32619666680309742499921124558, 2.65359584894682687960095469669, 4.16131195560831049601046063252, 4.63062737316110022346372612990, 5.57926622448507467137804352598, 6.48019793837594636385020945355, 7.09580040383556495339742141677, 7.88399086813646708417462037298, 9.177400825074426260287499990765, 9.577655384678036640036263408194

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