↑

Properties

Label 2-12e3-9.7-c1-0-4
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: $\chi_{1728} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.095055311\)
\(L(\frac12)\) \(\approx\) \(1.095055311\)
\(L(\frac{3}{2})\) 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

−9.449897543139573602730590175679, −8.555171415201132661796547942926, −7.69868662787631948015183337647, −7.41226332453920547113127767791, −5.86217215337653027325673199184, −5.18375580261884796647898516532, −4.54980701584794822986163276050, −3.53632459762643433209512507899, −2.30359926762647347589050939226, −0.902145862748046553481559302811, 0.56064554387638234113903828856, 2.63061474922682986486377796578, 3.26081652406158026469231423218, 3.99290492740254117696090056014, 5.19971120975202089336059860685, 6.39344051664995901418091682841, 6.81024313949086209439192751563, 7.74904203039847447921191639937, 8.161913928042657407722284873963, 9.379593379424966742521657181951

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