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Properties

Label 2-12e3-9.2-c2-0-24
Degree $2$
Conductor $1728$
Sign $0.939 - 0.342i$
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 − 1.73i)5-s + (1 − 1.73i)7-s + (1.5 + 0.866i)11-s + (−2 − 3.46i)13-s + 15.5i·17-s + 11·19-s + (−24 + 13.8i)23-s + (−6.5 + 11.2i)25-s + (39 + 22.5i)29-s + (16 + 27.7i)31-s − 6.92i·35-s + 34·37-s + (10.5 − 6.06i)41-s + (30.5 − 52.8i)43-s + (−42 − 24.2i)47-s + ⋯
L(s)  = 1  + (0.600 − 0.346i)5-s + (0.142 − 0.247i)7-s + (0.136 + 0.0787i)11-s + (−0.153 − 0.266i)13-s + 0.916i·17-s + 0.578·19-s + (−1.04 + 0.602i)23-s + (−0.260 + 0.450i)25-s + (1.34 + 0.776i)29-s + (0.516 + 0.893i)31-s − 0.197i·35-s + 0.918·37-s + (0.256 − 0.147i)41-s + (0.709 − 1.22i)43-s + (−0.893 − 0.515i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.939 - 0.342i$
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: \((2,\ 1728,\ (\ :1),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.294535119\)
\(L(\frac12)\) \(\approx\) \(2.294535119\)
\(L(2)\) 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 \( 1 + (-3 + 1.73i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 15.5iT - 289T^{2} \)
19 \( 1 - 11T + 361T^{2} \)
23 \( 1 + (24 - 13.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-39 - 22.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-16 - 27.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 34T + 1.36e3T^{2} \)
41 \( 1 + (-10.5 + 6.06i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-30.5 + 52.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (42 + 24.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (43.5 - 25.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-28 + 48.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 31.1iT - 5.04e3T^{2} \)
73 \( 1 - 65T + 5.32e3T^{2} \)
79 \( 1 + (-19 + 32.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-42 - 24.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 124. iT - 7.92e3T^{2} \)
97 \( 1 + (-57.5 + 99.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.222003443152315735949864415568, −8.391228727202896243085955318368, −7.68062651974702613036548314241, −6.74080540429973576334045933050, −5.88491713042549698762462270742, −5.17820563449725555935489719178, −4.21146592520024317955979405584, −3.22650144128127726678644209095, −1.98538916157057060863501078713, −1.01597025968342684514660818649, 0.72494137192645366615073850741, 2.18384126598263686343941782334, 2.84733732759649112125484188750, 4.18632115818431877416379360668, 4.95531022690702713137072996360, 6.10870110921482145963700700772, 6.42662133696954444876064283059, 7.62736426483450422907787660881, 8.199626826057060386833082768386, 9.294480692191545555815588915252

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