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Properties

Label 2-12e3-9.7-c1-0-15
Degree $2$
Conductor $1728$
Sign $0.948 + 0.315i$
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  + (1.68 + 2.92i)5-s + (2.35 − 4.07i)7-s + (0.437 − 0.758i)11-s + (−0.686 − 1.18i)13-s + 2.37·17-s + 5.57·19-s + (−2.35 − 4.07i)23-s + (−3.18 + 5.51i)25-s + (2.68 − 4.65i)29-s + (−3.22 − 5.58i)31-s + 15.8·35-s − 4·37-s + (0.5 + 0.866i)41-s + (0.437 − 0.758i)43-s + (−2.35 + 4.07i)47-s + ⋯
L(s)  = 1  + (0.754 + 1.30i)5-s + (0.888 − 1.53i)7-s + (0.131 − 0.228i)11-s + (−0.190 − 0.329i)13-s + 0.575·17-s + 1.27·19-s + (−0.490 − 0.849i)23-s + (−0.637 + 1.10i)25-s + (0.498 − 0.863i)29-s + (−0.579 − 1.00i)31-s + 2.68·35-s − 0.657·37-s + (0.0780 + 0.135i)41-s + (0.0667 − 0.115i)43-s + (−0.342 + 0.594i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.948 + 0.315i$
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: \((2,\ 1728,\ (\ :1/2),\ 0.948 + 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.272859878\)
\(L(\frac12)\) \(\approx\) \(2.272859878\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 2.92i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.35 + 4.07i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.437 + 0.758i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.686 + 1.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.37T + 17T^{2} \)
19 \( 1 - 5.57T + 19T^{2} \)
23 \( 1 + (2.35 + 4.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.68 + 4.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.22 + 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.437 + 0.758i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.35 - 4.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (-4.26 - 7.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.05 - 1.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.26 - 7.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.40T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + (-3.22 + 5.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.47 - 2.55i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.677331633457423816034170067317, −8.240458954045576066450745984237, −7.58231835298547816023466935662, −6.99772012703415784671520370775, −6.15579821080933653052588598915, −5.26165375593498213170447788368, −4.17720011360168402448972740891, −3.31093499798757949900716491875, −2.25445038957899373445358037708, −0.974661062406748184115686956954, 1.37976285560405631397523295244, 2.01254726180545942120668644812, 3.34744672942999697510313345598, 4.83754292557802924955746539847, 5.25916384397208404247625418021, 5.75272172242009526911873304024, 6.97587617603862873374609325557, 8.117289454129547368849617803872, 8.569576330474442694237740031511, 9.477089785860486166250423933701

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