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Properties

Label 2-12e3-36.7-c2-0-8
Degree $2$
Conductor $1728$
Sign $0.889 - 0.457i$
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  + (−4.03 − 6.98i)5-s + (−3.90 − 2.25i)7-s + (3.25 + 1.88i)11-s + (3.52 + 6.10i)13-s − 0.517·17-s + 16.4i·19-s + (−27.7 + 15.9i)23-s + (−19.9 + 34.6i)25-s + (9.48 − 16.4i)29-s + (−13.1 + 7.58i)31-s + 36.3i·35-s − 0.592·37-s + (−12.3 − 21.4i)41-s + (−27.8 − 16.0i)43-s + (52.4 + 30.2i)47-s + ⋯
L(s)  = 1  + (−0.806 − 1.39i)5-s + (−0.557 − 0.321i)7-s + (0.296 + 0.171i)11-s + (0.271 + 0.469i)13-s − 0.0304·17-s + 0.864i·19-s + (−1.20 + 0.695i)23-s + (−0.799 + 1.38i)25-s + (0.327 − 0.566i)29-s + (−0.423 + 0.244i)31-s + 1.03i·35-s − 0.0160·37-s + (−0.301 − 0.522i)41-s + (−0.648 − 0.374i)43-s + (1.11 + 0.644i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.889 - 0.457i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.889 - 0.457i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.024631662\)
\(L(\frac12)\) \(\approx\) \(1.024631662\)
\(L(2)\) 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 \( 1 + (4.03 + 6.98i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (3.90 + 2.25i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.25 - 1.88i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.52 - 6.10i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 0.517T + 289T^{2} \)
19 \( 1 - 16.4iT - 361T^{2} \)
23 \( 1 + (27.7 - 15.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-9.48 + 16.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (13.1 - 7.58i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 0.592T + 1.36e3T^{2} \)
41 \( 1 + (12.3 + 21.4i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (27.8 + 16.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-52.4 - 30.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 0.664T + 2.80e3T^{2} \)
59 \( 1 + (30.5 - 17.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.7 - 58.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-74.4 + 42.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 56.4iT - 5.04e3T^{2} \)
73 \( 1 - 131.T + 5.32e3T^{2} \)
79 \( 1 + (-126. - 73.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-87.1 - 50.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 25.8T + 7.92e3T^{2} \)
97 \( 1 + (48.2 - 83.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.211204441670875274236193056209, −8.331391140754931059811356032635, −7.82387972524210188526847631298, −6.85487614386148611147223135990, −5.90978459638589281216818707245, −5.02294791646476912910717922391, −4.03088564218269745080483238727, −3.65929833664332523058617980724, −1.92184264970924322566090167002, −0.812561919857023008717115823358, 0.37201140736744352305994127879, 2.25244322591372208870160746166, 3.17946773001280951010946734768, 3.75993135930902746900081703146, 4.91833185628425649808469853153, 6.22568687960086011075695962348, 6.54809992037123900527039915775, 7.48631348919235989836160657130, 8.157535912577391182429568846963, 9.074436665091318160057857914785

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