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Properties

Label 2-12e3-24.11-c3-0-32
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  + 11.9·5-s + 29.7i·7-s + 41.2i·11-s + 17.8·25-s + 218.·29-s + 257. i·31-s + 355. i·35-s − 540.·49-s + 247.·53-s + 493. i·55-s + 717. i·59-s − 1.20e3·73-s − 1.22e3·77-s − 1.37e3i·79-s − 1.50e3i·83-s + ⋯
L(s)  = 1  + 1.06·5-s + 1.60i·7-s + 1.13i·11-s + 0.142·25-s + 1.39·29-s + 1.49i·31-s + 1.71i·35-s − 1.57·49-s + 0.641·53-s + 1.20i·55-s + 1.58i·59-s − 1.93·73-s − 1.81·77-s − 1.95i·79-s − 1.98i·83-s + ⋯

Functional equation

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

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: $\chi_{1728} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :3/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.237381287\)
\(L(\frac12)\) \(\approx\) \(2.237381287\)
\(L(\frac{5}{2})\) 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 \( 1 - 11.9T + 125T^{2} \)
7 \( 1 - 29.7iT - 343T^{2} \)
11 \( 1 - 41.2iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 218.T + 2.43e4T^{2} \)
31 \( 1 - 257. iT - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 + 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 247.T + 1.48e5T^{2} \)
59 \( 1 - 717. iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 + 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 1.20e3T + 3.89e5T^{2} \)
79 \( 1 + 1.37e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.50e3iT - 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 - 1.86e3T + 9.12e5T^{2} \)
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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

−9.000802412273071357715585113551, −8.886341526883864031317984750452, −7.68614421233544601076035659772, −6.68255462868458071047207586836, −5.99661357197809747349082324846, −5.27107885784735038676736280425, −4.54861003030928335441621835242, −3.00537923533238538888345473837, −2.26795281373470640365500841564, −1.47240105764935653483230976916, 0.46315754505893177143653726586, 1.30513344451521994205020323121, 2.55518823318183839013848617267, 3.62738071969254808291772780691, 4.44958981862286318543380856090, 5.52524926812725012715193068207, 6.26488295494164422305118348454, 6.97384756574072564500451090786, 7.900857933053316978598257500041, 8.617499529156901808084127026481

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