Properties

Label 2-12e3-8.3-c2-0-25
Degree $2$
Conductor $1728$
Sign $0.965 - 0.258i$
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  − 3.99i·5-s + 7.74i·7-s − 7.68·11-s − 1.42i·13-s + 22.3·17-s − 21.9·19-s − 18.1i·23-s + 9.02·25-s − 15.2i·29-s − 10.2i·31-s + 30.9·35-s + 59.1i·37-s + 41.3·41-s − 11.0·43-s + 63.6i·47-s + ⋯
L(s)  = 1  − 0.799i·5-s + 1.10i·7-s − 0.699·11-s − 0.109i·13-s + 1.31·17-s − 1.15·19-s − 0.789i·23-s + 0.360·25-s − 0.527i·29-s − 0.329i·31-s + 0.884·35-s + 1.59i·37-s + 1.00·41-s − 0.255·43-s + 1.35i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.965 - 0.258i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.965 - 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.800153362\)
\(L(\frac12)\) \(\approx\) \(1.800153362\)
\(L(2)\) 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 \( 1 + 3.99iT - 25T^{2} \)
7 \( 1 - 7.74iT - 49T^{2} \)
11 \( 1 + 7.68T + 121T^{2} \)
13 \( 1 + 1.42iT - 169T^{2} \)
17 \( 1 - 22.3T + 289T^{2} \)
19 \( 1 + 21.9T + 361T^{2} \)
23 \( 1 + 18.1iT - 529T^{2} \)
29 \( 1 + 15.2iT - 841T^{2} \)
31 \( 1 + 10.2iT - 961T^{2} \)
37 \( 1 - 59.1iT - 1.36e3T^{2} \)
41 \( 1 - 41.3T + 1.68e3T^{2} \)
43 \( 1 + 11.0T + 1.84e3T^{2} \)
47 \( 1 - 63.6iT - 2.20e3T^{2} \)
53 \( 1 + 19.2iT - 2.80e3T^{2} \)
59 \( 1 + 16.1T + 3.48e3T^{2} \)
61 \( 1 + 27.7iT - 3.72e3T^{2} \)
67 \( 1 - 60.6T + 4.48e3T^{2} \)
71 \( 1 - 134. iT - 5.04e3T^{2} \)
73 \( 1 - 90.0T + 5.32e3T^{2} \)
79 \( 1 + 12.0iT - 6.24e3T^{2} \)
83 \( 1 - 164.T + 6.88e3T^{2} \)
89 \( 1 - 35.6T + 7.92e3T^{2} \)
97 \( 1 + 147.T + 9.40e3T^{2} \)
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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

−9.122056198717350216861208590015, −8.254428364148105833563308049414, −7.987559981392179807755555443170, −6.63120732150041977706652527719, −5.81924151200608844452550436327, −5.13273725531366158851384628458, −4.35084371016725557176455228035, −3.03375404025234203456353321004, −2.17307358079332710620423906057, −0.822609075256474273486912736573, 0.65953633221048804331524976978, 2.06289048567130608104652228958, 3.23344959811161317001909981309, 3.91166332920124436154935485466, 5.01420169228185993674822809169, 5.92387616805120183548946050371, 6.89319654799916547596962550944, 7.43877625184809259453494970821, 8.113883073908815194823484457926, 9.187037496131133211882337249929

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