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Properties

Label 2-12e3-36.23-c1-0-21
Degree $2$
Conductor $1728$
Sign $-0.999 + 0.0281i$
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  + (−2.18 − 1.26i)5-s + (1.10 − 0.637i)7-s + (0.252 + 0.437i)11-s + (−1.18 + 2.05i)13-s − 0.792i·17-s − 4.70i·19-s + (−1.61 + 2.78i)23-s + (0.686 + 1.18i)25-s + (2.18 − 1.26i)29-s + (−7.04 − 4.06i)31-s − 3.22·35-s + 6.74·37-s + (−5.87 − 3.39i)41-s + (−6.69 + 3.86i)43-s + (−0.599 − 1.03i)47-s + ⋯
L(s)  = 1  + (−0.977 − 0.564i)5-s + (0.417 − 0.241i)7-s + (0.0761 + 0.131i)11-s + (−0.328 + 0.569i)13-s − 0.192i·17-s − 1.07i·19-s + (−0.335 + 0.581i)23-s + (0.137 + 0.237i)25-s + (0.405 − 0.234i)29-s + (−1.26 − 0.730i)31-s − 0.544·35-s + 1.10·37-s + (−0.917 − 0.529i)41-s + (−1.02 + 0.589i)43-s + (−0.0874 − 0.151i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.999 + 0.0281i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.999 + 0.0281i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2895064418\)
\(L(\frac12)\) \(\approx\) \(0.2895064418\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 1.26i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.10 + 0.637i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.252 - 0.437i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.18 - 2.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.792iT - 17T^{2} \)
19 \( 1 + 4.70iT - 19T^{2} \)
23 \( 1 + (1.61 - 2.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.18 + 1.26i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.04 + 4.06i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.74T + 37T^{2} \)
41 \( 1 + (5.87 + 3.39i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.69 - 3.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.599 + 1.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.87iT - 53T^{2} \)
59 \( 1 + (6.18 - 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.18 + 2.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.69 + 3.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 3.37T + 73T^{2} \)
79 \( 1 + (8.55 - 4.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.82 + 6.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 + (5.24 + 9.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.949140693604443169817036298349, −8.027985585885420330952644332907, −7.49980686070545875485666132097, −6.69128875755532765240015634742, −5.54515641143248990029478983589, −4.55727420538708081383874238610, −4.14856433317200854224632458410, −2.90677093891733064306820097722, −1.55765566820182753069189586846, −0.10876514460939150405045194650, 1.68099078736303798302124313777, 3.02408282208190283879432884505, 3.74098737810386314294192071421, 4.74484373743841642221673856737, 5.66307084681721244406792658424, 6.61046733580009729006411246255, 7.46116458852352820090716021485, 8.098670420981455350167004349564, 8.686452277258263391991833990565, 9.847294797783704298749674445927

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