Properties

Label 2-12e3-9.4-c1-0-2
Degree $2$
Conductor $1728$
Sign $-0.939 - 0.342i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 3.46i)5-s + (1 + 1.73i)7-s + (2.5 + 4.33i)11-s + (−1 + 1.73i)13-s + 3·17-s − 19-s + (−3 + 5.19i)23-s + (−5.49 − 9.52i)25-s + (1 + 1.73i)29-s + (2 − 3.46i)31-s − 7.99·35-s + 8·37-s + (0.5 − 0.866i)41-s + (−3.5 − 6.06i)43-s + (1 + 1.73i)47-s + ⋯
L(s)  = 1  + (−0.894 + 1.54i)5-s + (0.377 + 0.654i)7-s + (0.753 + 1.30i)11-s + (−0.277 + 0.480i)13-s + 0.727·17-s − 0.229·19-s + (−0.625 + 1.08i)23-s + (−1.09 − 1.90i)25-s + (0.185 + 0.321i)29-s + (0.359 − 0.622i)31-s − 1.35·35-s + 1.31·37-s + (0.0780 − 0.135i)41-s + (−0.533 − 0.924i)43-s + (0.145 + 0.252i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.939 - 0.342i$
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: \((2,\ 1728,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.251900673\)
\(L(\frac12)\) \(\approx\) \(1.251900673\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.807300617944381312571083850578, −8.907051314615149302917786750051, −7.78487844638075735792355294607, −7.38818885556590086188628219527, −6.62053675617580784613380427019, −5.78729004716085054148508240714, −4.51814910382253906156582526099, −3.81659034575436247469263260460, −2.78588164234891155737582409680, −1.83255441510461586525611563197, 0.52259469472536589472320498158, 1.29344922144440444331405783014, 3.13080301206242063084309469068, 4.12430883991265102219677497826, 4.64447937497497239643942116314, 5.62438557068590011087425888019, 6.50996204264150024609152054583, 7.84517494682906175480035352652, 8.064057370420101004159721419440, 8.811240910489263498153362467917

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