Properties

Label 2-12e3-8.5-c3-0-80
Degree $2$
Conductor $1728$
Sign $-0.707 + 0.707i$
Analytic cond. $101.955$
Root an. cond. $10.0972$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16.7i·5-s + 22.3·7-s + 51.9i·11-s − 28.0i·13-s − 60.9·17-s + 162. i·19-s − 83.2·23-s − 156.·25-s − 217. i·29-s + 174.·31-s − 374. i·35-s − 407. i·37-s + 196.·41-s − 345. i·43-s − 335.·47-s + ⋯
L(s)  = 1  − 1.50i·5-s + 1.20·7-s + 1.42i·11-s − 0.598i·13-s − 0.869·17-s + 1.96i·19-s − 0.755·23-s − 1.25·25-s − 1.39i·29-s + 1.00·31-s − 1.80i·35-s − 1.81i·37-s + 0.748·41-s − 1.22i·43-s − 1.04·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(101.955\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.568820488\)
\(L(\frac12)\) \(\approx\) \(1.568820488\)
\(L(\frac{5}{2})\) 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 \( 1 + 16.7iT - 125T^{2} \)
7 \( 1 - 22.3T + 343T^{2} \)
11 \( 1 - 51.9iT - 1.33e3T^{2} \)
13 \( 1 + 28.0iT - 2.19e3T^{2} \)
17 \( 1 + 60.9T + 4.91e3T^{2} \)
19 \( 1 - 162. iT - 6.85e3T^{2} \)
23 \( 1 + 83.2T + 1.21e4T^{2} \)
29 \( 1 + 217. iT - 2.43e4T^{2} \)
31 \( 1 - 174.T + 2.97e4T^{2} \)
37 \( 1 + 407. iT - 5.06e4T^{2} \)
41 \( 1 - 196.T + 6.89e4T^{2} \)
43 \( 1 + 345. iT - 7.95e4T^{2} \)
47 \( 1 + 335.T + 1.03e5T^{2} \)
53 \( 1 + 234. iT - 1.48e5T^{2} \)
59 \( 1 - 321. iT - 2.05e5T^{2} \)
61 \( 1 + 739. iT - 2.26e5T^{2} \)
67 \( 1 - 20iT - 3.00e5T^{2} \)
71 \( 1 + 418.T + 3.57e5T^{2} \)
73 \( 1 - 397.T + 3.89e5T^{2} \)
79 \( 1 - 95.6T + 4.93e5T^{2} \)
83 \( 1 + 348. iT - 5.71e5T^{2} \)
89 \( 1 + 1.49e3T + 7.04e5T^{2} \)
97 \( 1 - 881.T + 9.12e5T^{2} \)
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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

−8.371873047859299910574383365648, −8.133966252131542695963765584616, −7.34269892797669480424662482124, −6.00035676870645904061391680198, −5.29485838048465511551657735609, −4.46057844846389312663936233206, −4.04617159936699514143749145994, −2.10876787636461743366480871122, −1.59773549971785956342591427238, −0.33084168405482716410499986269, 1.21353645359614542259784971800, 2.51761089074934207145791874072, 3.09856868487461781085138774510, 4.32759097917518276940869523885, 5.10520769293795865459731224702, 6.38341926413017309614126204624, 6.67126984778837477398765968903, 7.67709510499580151410599949264, 8.440829761449917109405645185963, 9.118748771422194369204924786171

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