Properties

Label 2-12e3-9.5-c2-0-19
Degree $2$
Conductor $1728$
Sign $0.766 - 0.642i$
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  + (6.39 + 3.69i)5-s + (−3.39 − 5.88i)7-s + (−5.29 + 3.05i)11-s + (8.39 − 14.5i)13-s + 25.1i·17-s − 17.5·19-s + (12.3 + 7.15i)23-s + (14.7 + 25.6i)25-s + (16.1 − 9.35i)29-s + (−23.3 + 40.5i)31-s − 50.2i·35-s + 49.5·37-s + (34.5 + 19.9i)41-s + (22.0 + 38.2i)43-s + (28.8 − 16.6i)47-s + ⋯
L(s)  = 1  + (1.27 + 0.738i)5-s + (−0.485 − 0.841i)7-s + (−0.481 + 0.278i)11-s + (0.646 − 1.11i)13-s + 1.48i·17-s − 0.926·19-s + (0.539 + 0.311i)23-s + (0.591 + 1.02i)25-s + (0.558 − 0.322i)29-s + (−0.754 + 1.30i)31-s − 1.43i·35-s + 1.34·37-s + (0.841 + 0.485i)41-s + (0.513 + 0.890i)43-s + (0.612 − 0.353i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.766 - 0.642i$
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: \((2,\ 1728,\ (\ :1),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.375955434\)
\(L(\frac12)\) \(\approx\) \(2.375955434\)
\(L(2)\) 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 \( 1 + (-6.39 - 3.69i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (3.39 + 5.88i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (5.29 - 3.05i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.39 + 14.5i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 25.1iT - 289T^{2} \)
19 \( 1 + 17.5T + 361T^{2} \)
23 \( 1 + (-12.3 - 7.15i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-16.1 + 9.35i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (23.3 - 40.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 49.5T + 1.36e3T^{2} \)
41 \( 1 + (-34.5 - 19.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-22.0 - 38.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-28.8 + 16.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 10.1iT - 2.80e3T^{2} \)
59 \( 1 + (-14.2 - 8.25i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-10.6 - 18.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-43.4 + 75.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 30.2iT - 5.04e3T^{2} \)
73 \( 1 + 48.7T + 5.32e3T^{2} \)
79 \( 1 + (-55.7 - 96.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-85.0 + 49.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 75.5iT - 7.92e3T^{2} \)
97 \( 1 + (-70.2 - 121. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.409128055118938965738387392697, −8.398573045145746832308717464942, −7.61305629842975669307550116613, −6.60690204339630766507547119173, −6.15179433077466584368758205719, −5.35570672241337315871028624429, −4.10573286812878458859996317913, −3.17590317597672311108422397440, −2.24894719206321869406560771535, −1.04478869941113221053507293258, 0.71702980053561517363961142947, 2.09463164723989703947101496843, 2.68896821245819096831910894596, 4.17633513110979068804825731805, 5.09504457202764020781060325587, 5.86596597334672272893419842730, 6.37151299657640513641369047999, 7.41879478383430594777320018942, 8.611693480231682762190552936627, 9.224454548053187375963853369897

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