Properties

Label 2-12e3-8.5-c1-0-30
Degree $2$
Conductor $1728$
Sign $-0.965 - 0.258i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46i·5-s − 1.73·7-s − 6i·11-s + 5.19i·13-s − 6·17-s − 5i·19-s + 3.46·23-s − 6.99·25-s + 6.92i·29-s − 3.46·31-s + 5.99i·35-s + 1.73i·37-s − 4i·43-s + 3.46·47-s − 4·49-s + ⋯
L(s)  = 1  − 1.54i·5-s − 0.654·7-s − 1.80i·11-s + 1.44i·13-s − 1.45·17-s − 1.14i·19-s + 0.722·23-s − 1.39·25-s + 1.28i·29-s − 0.622·31-s + 1.01i·35-s + 0.284i·37-s − 0.609i·43-s + 0.505·47-s − 0.571·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.965 - 0.258i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.965 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5668625877\)
\(L(\frac12)\) \(\approx\) \(0.5668625877\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 - 1.73iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 6.92iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 12.1iT - 61T^{2} \)
67 \( 1 - 5iT - 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 5.19T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−8.952774364612544803326093142608, −8.599206564057439879624154011392, −7.18173200956071187532436414742, −6.48740459502075039145385782136, −5.56503337399670877144535030039, −4.74490439064127255540267277504, −3.97351412925969355109850596586, −2.82605179923579814619731568052, −1.41474294288857573765650224461, −0.20885354321755968706976410599, 2.04252804209525565536207341093, 2.84561326542897479118525896008, 3.74344345731746729832954448024, 4.76150642500062643971533485193, 5.99096607040410814367549485677, 6.57263687288255213501342879076, 7.37689922160531482214435268109, 7.84416165172247766805614126758, 9.166651964715582920921978562681, 9.941949001368524104314474968925

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