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Properties

Label 2-12e3-9.2-c2-0-31
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: $\chi_{1728} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.254 + 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.375978693\)
\(L(\frac12)\) \(\approx\) \(1.375978693\)
\(L(2)\) 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

−8.792730113107604990490280887143, −7.997812142217493435198198793595, −7.42381679591573448900452535091, −6.87714036248286677726104351300, −5.65650573719261387421786866911, −4.75913808368800199163275959222, −3.79305782923843292694516225059, −3.23410265210801663755017448517, −1.69997281667582606757403380750, −0.43895845157770047416594616277, 1.01617525367617213605359770311, 2.32530129951863677873215299370, 3.33984548611089662609378869030, 4.55321340516835165721318229061, 5.02592661951183376785039474602, 5.97209785098330548679517430448, 7.11755125655007339696744012332, 7.76950438403268358821094918332, 8.526987527571349532171453410935, 9.204303499763623176712385429718

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