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Properties

Label 2-12e3-36.31-c2-0-3
Degree $2$
Conductor $1728$
Sign $-0.967 + 0.252i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.454 + 0.787i)5-s + (−6.10 + 3.52i)7-s + (−6.96 + 4.02i)11-s + (−3.35 + 5.81i)13-s + 26.3·17-s − 20.5i·19-s + (21.8 + 12.6i)23-s + (12.0 + 20.9i)25-s + (15.1 + 26.2i)29-s + (−0.120 − 0.0693i)31-s − 6.41i·35-s − 69.7·37-s + (29.3 − 50.8i)41-s + (−2.45 + 1.41i)43-s + (−70.7 + 40.8i)47-s + ⋯
L(s)  = 1  + (−0.0909 + 0.157i)5-s + (−0.872 + 0.503i)7-s + (−0.633 + 0.365i)11-s + (−0.258 + 0.447i)13-s + 1.54·17-s − 1.08i·19-s + (0.949 + 0.547i)23-s + (0.483 + 0.837i)25-s + (0.523 + 0.906i)29-s + (−0.00387 − 0.00223i)31-s − 0.183i·35-s − 1.88·37-s + (0.716 − 1.24i)41-s + (−0.0571 + 0.0330i)43-s + (−1.50 + 0.869i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.967 + 0.252i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.967 + 0.252i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2358416317\)
\(L(\frac12)\) \(\approx\) \(0.2358416317\)
\(L(2)\) 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 \( 1 + (0.454 - 0.787i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (6.10 - 3.52i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (6.96 - 4.02i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.35 - 5.81i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 26.3T + 289T^{2} \)
19 \( 1 + 20.5iT - 361T^{2} \)
23 \( 1 + (-21.8 - 12.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-15.1 - 26.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (0.120 + 0.0693i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 69.7T + 1.36e3T^{2} \)
41 \( 1 + (-29.3 + 50.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (2.45 - 1.41i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (70.7 - 40.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 30.0T + 2.80e3T^{2} \)
59 \( 1 + (77.1 + 44.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (24.0 + 41.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (44.0 + 25.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 68.4iT - 5.04e3T^{2} \)
73 \( 1 + 22.1T + 5.32e3T^{2} \)
79 \( 1 + (34.4 - 19.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (23.0 - 13.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 25.7T + 7.92e3T^{2} \)
97 \( 1 + (52.3 + 90.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.340165297785044307463781393434, −9.084209492483043750396202086718, −7.86001438344937745774283971207, −7.15947801119385698039556177536, −6.48178250252367657355768673318, −5.37303881120328646616831918560, −4.86401208528824625594627968185, −3.34230857948160390884356900099, −2.93743834435003252649324578005, −1.51916620324430889346853008534, 0.06636453876717912340577243844, 1.21278326629309510341090974873, 2.85578083957115533896955190482, 3.42019807464164893340684103275, 4.56948075395261318636247774091, 5.50126439238352185463669394277, 6.26359040403753342305973748481, 7.15840167966029776570140472695, 7.982190603470724270648464155919, 8.547691940236501481167198831897

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