Properties

Label 2-12e3-8.3-c2-0-22
Degree $2$
Conductor $1728$
Sign $0.707 - 0.707i$
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  + 3.16i·5-s − 3.48i·7-s − 1.13·11-s + 14.9·25-s + 29.3i·29-s − 23.4i·31-s + 11.0·35-s + 36.8·49-s + 56.9i·53-s − 3.60i·55-s + 117.·59-s − 93.7·73-s + 3.96i·77-s + 58i·79-s + 67.0·83-s + ⋯
L(s)  = 1  + 0.633i·5-s − 0.497i·7-s − 0.103·11-s + 0.598·25-s + 1.01i·29-s − 0.755i·31-s + 0.315·35-s + 0.752·49-s + 1.07i·53-s − 0.0655i·55-s + 1.99·59-s − 1.28·73-s + 0.0514i·77-s + 0.734i·79-s + 0.807·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(47.0845\)
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: \((2,\ 1728,\ (\ :1),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.838251150\)
\(L(\frac12)\) \(\approx\) \(1.838251150\)
\(L(2)\) 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 \( 1 - 3.16iT - 25T^{2} \)
7 \( 1 + 3.48iT - 49T^{2} \)
11 \( 1 + 1.13T + 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 - 29.3iT - 841T^{2} \)
31 \( 1 + 23.4iT - 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 - 56.9iT - 2.80e3T^{2} \)
59 \( 1 - 117.T + 3.48e3T^{2} \)
61 \( 1 - 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 93.7T + 5.32e3T^{2} \)
79 \( 1 - 58iT - 6.24e3T^{2} \)
83 \( 1 - 67.0T + 6.88e3T^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 - 61.0T + 9.40e3T^{2} \)
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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

−9.232183727983189895237436549811, −8.453936475589109492066062377269, −7.48385846669784161228249480805, −6.96462310184497491668739112505, −6.09938031891792078810915343859, −5.15584508502648628850107060563, −4.16494585156782171368434180698, −3.27409818666700371604734190803, −2.29988315488131069726556744327, −0.922920546563537141627362439046, 0.61806009169192532007274278592, 1.89968797854150380301015951127, 2.96983775772443191495723507365, 4.09967948891198661073979496537, 4.99353900038615228335703567306, 5.69239366974167925944207639758, 6.61579215831667185599300182210, 7.50708844816533753023217874979, 8.456766620751209954326238199508, 8.872187373305128112308119364235

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