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Properties

Label 2-12e3-9.5-c2-0-24
Degree $2$
Conductor $1728$
Sign $0.999 + 0.00316i$
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  + (3.32 + 1.91i)5-s + (−2.70 − 4.68i)7-s + (10.4 − 6.01i)11-s + (0.140 − 0.242i)13-s + 22.7i·17-s + 2.05·19-s + (26.8 + 15.5i)23-s + (−5.13 − 8.90i)25-s + (−4.59 + 2.65i)29-s + (9.81 − 17.0i)31-s − 20.7i·35-s − 65.7·37-s + (59.1 + 34.1i)41-s + (28.1 + 48.7i)43-s + (3.83 − 2.21i)47-s + ⋯
L(s)  = 1  + (0.664 + 0.383i)5-s + (−0.386 − 0.669i)7-s + (0.946 − 0.546i)11-s + (0.0107 − 0.0186i)13-s + 1.33i·17-s + 0.108·19-s + (1.16 + 0.674i)23-s + (−0.205 − 0.356i)25-s + (−0.158 + 0.0914i)29-s + (0.316 − 0.548i)31-s − 0.593i·35-s − 1.77·37-s + (1.44 + 0.832i)41-s + (0.653 + 1.13i)43-s + (0.0816 − 0.0471i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00316i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.999 + 0.00316i$
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: \((2,\ 1728,\ (\ :1),\ 0.999 + 0.00316i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.396944654\)
\(L(\frac12)\) \(\approx\) \(2.396944654\)
\(L(2)\) 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 \( 1 + (-3.32 - 1.91i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (2.70 + 4.68i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-10.4 + 6.01i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.140 + 0.242i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 22.7iT - 289T^{2} \)
19 \( 1 - 2.05T + 361T^{2} \)
23 \( 1 + (-26.8 - 15.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (4.59 - 2.65i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-9.81 + 17.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 65.7T + 1.36e3T^{2} \)
41 \( 1 + (-59.1 - 34.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-28.1 - 48.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-3.83 + 2.21i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 34.9iT - 2.80e3T^{2} \)
59 \( 1 + (-23.4 - 13.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (51.7 + 89.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.3 + 68.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 94.2iT - 5.04e3T^{2} \)
73 \( 1 - 126.T + 5.32e3T^{2} \)
79 \( 1 + (-45.2 - 78.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-45.1 + 26.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 72.1iT - 7.92e3T^{2} \)
97 \( 1 + (9.38 + 16.2i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.317602909849730116007118143428, −8.367615661527928887215338776652, −7.50807575338312565696353044645, −6.47015665288375513690164485845, −6.23075028423287018529733653921, −5.09857455059982140850049407504, −3.92037090810657167411277686248, −3.30226676442444843803125173950, −1.98600968888808472891533551923, −0.884296461151362242058841717512, 0.877434763374457683336608573817, 2.08322713642387082882208336456, 3.01095865362444574600061981596, 4.20859114739143936876911183842, 5.18416088143075913142846165359, 5.78185280031056980689718883422, 6.84507297950566010858793834588, 7.30578306267673915069983171748, 8.762235673481018626381800059892, 9.083593925450747474407177107124

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