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Properties

Label 2-12e3-9.4-c1-0-11
Degree $2$
Conductor $1728$
Sign $0.996 - 0.0825i$
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  + (0.5 − 0.866i)5-s + (1.72 + 2.98i)7-s + (0.724 + 1.25i)11-s + (2.94 − 5.10i)13-s − 4.89·17-s + 4·19-s + (2.72 − 4.71i)23-s + (2 + 3.46i)25-s + (−0.0505 − 0.0874i)29-s + (1.27 − 2.20i)31-s + 3.44·35-s + 0.898·37-s + (−5.94 + 10.3i)41-s + (1.17 + 2.03i)43-s + (3.17 + 5.49i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.651 + 1.12i)7-s + (0.218 + 0.378i)11-s + (0.818 − 1.41i)13-s − 1.18·17-s + 0.917·19-s + (0.568 − 0.984i)23-s + (0.400 + 0.692i)25-s + (−0.00937 − 0.0162i)29-s + (0.229 − 0.396i)31-s + 0.583·35-s + 0.147·37-s + (−0.929 + 1.60i)41-s + (0.179 + 0.310i)43-s + (0.463 + 0.801i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.996 - 0.0825i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.996 - 0.0825i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.065998170\)
\(L(\frac12)\) \(\approx\) \(2.065998170\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.72 - 2.98i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.724 - 1.25i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.94 + 5.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-2.72 + 4.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0505 + 0.0874i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.27 + 2.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.898T + 37T^{2} \)
41 \( 1 + (5.94 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.17 - 2.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.17 - 5.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.89T + 53T^{2} \)
59 \( 1 + (-7.17 + 12.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.94 + 6.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.17 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.79T + 71T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 + (-6.72 - 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.275 + 0.476i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (-1.94 - 3.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.192549985877580440630255961023, −8.535526474325837693306333513048, −8.014348212281432522867110797302, −6.88701627700647538406160476821, −5.97361148878903154157960562778, −5.24634318619143262840983201375, −4.58423533164180212757894368288, −3.23574429797695283510368643474, −2.29480020105080556231685410458, −1.07336079612621537608805950955, 1.04951655921322591573620054157, 2.13356210915103850722693014870, 3.55066214829680230130596966838, 4.21386504088834079915415819672, 5.14450345167735074413660997233, 6.27139523363594037707538634087, 6.98654862241536531562171246045, 7.51363841545800908509403922792, 8.785171700080501159457487586262, 9.043556190990965701859040056762

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