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Properties

Label 2-12e3-9.5-c2-0-4
Degree $2$
Conductor $1728$
Sign $0.254 - 0.967i$
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  + (−4.5 − 2.59i)5-s + (−4.17 − 7.22i)7-s + (−0.825 + 0.476i)11-s + (−4.84 + 8.39i)13-s − 18.8i·17-s − 24.6·19-s + (0.825 + 0.476i)23-s + (1 + 1.73i)25-s + (11.8 − 6.84i)29-s + (1.52 − 2.63i)31-s + 43.3i·35-s − 46.6·37-s + (9.45 + 5.45i)41-s + (−22.5 − 39.0i)43-s + (39.2 − 22.6i)47-s + ⋯
L(s)  = 1  + (−0.900 − 0.519i)5-s + (−0.596 − 1.03i)7-s + (−0.0750 + 0.0433i)11-s + (−0.372 + 0.645i)13-s − 1.11i·17-s − 1.29·19-s + (0.0359 + 0.0207i)23-s + (0.0400 + 0.0692i)25-s + (0.408 − 0.235i)29-s + (0.0491 − 0.0850i)31-s + 1.23i·35-s − 1.26·37-s + (0.230 + 0.133i)41-s + (−0.523 − 0.907i)43-s + (0.834 − 0.481i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.254 - 0.967i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.254 - 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3370678710\)
\(L(\frac12)\) \(\approx\) \(0.3370678710\)
\(L(2)\) 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 \( 1 + (4.5 + 2.59i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (4.17 + 7.22i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (0.825 - 0.476i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.84 - 8.39i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 18.8iT - 289T^{2} \)
19 \( 1 + 24.6T + 361T^{2} \)
23 \( 1 + (-0.825 - 0.476i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-11.8 + 6.84i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-1.52 + 2.63i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 46.6T + 1.36e3T^{2} \)
41 \( 1 + (-9.45 - 5.45i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.5 + 39.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-39.2 + 22.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 94.3iT - 2.80e3T^{2} \)
59 \( 1 + (-16.2 - 9.39i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.54 - 11.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (37.5 - 64.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 18.0iT - 5.04e3T^{2} \)
73 \( 1 + 7.90T + 5.32e3T^{2} \)
79 \( 1 + (21.8 + 37.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-112. + 65.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + (-54.9 - 95.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.164748101692058090250245344982, −8.588191738071732057562102161765, −7.55863162453018401580623713611, −7.09199466576277378541847843617, −6.25412234960173953662990341393, −4.95775633334391059691668144819, −4.27521880927572824975039814065, −3.59898233839693262459428059056, −2.33162523459774573580619370478, −0.77902889776952302173268623478, 0.12264916664648274911283017166, 1.99625547958739637574804758579, 3.03413568149747838414691821612, 3.74839722905670773362314394198, 4.83985221859437776348368709027, 5.87363290908251210108568816817, 6.51746232615228626652799298551, 7.41042688604577072093136457250, 8.309666806972407349035336531104, 8.745180183627608722888019086204

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