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Properties

Label 2-12e3-24.5-c2-0-13
Degree $2$
Conductor $1728$
Sign $-0.258 - 0.965i$
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  + 7.74·5-s − 8.66·7-s − 13.4·11-s − 5.19i·13-s + 13.4i·17-s − 23i·19-s + 7.74i·23-s + 35.0·25-s + 30.9·29-s − 6.92·31-s − 67.0·35-s + 29.4i·37-s + 80.4i·41-s + 38i·43-s + 54.2i·47-s + ⋯
L(s)  = 1  + 1.54·5-s − 1.23·7-s − 1.21·11-s − 0.399i·13-s + 0.789i·17-s − 1.21i·19-s + 0.336i·23-s + 1.40·25-s + 1.06·29-s − 0.223·31-s − 1.91·35-s + 0.795i·37-s + 1.96i·41-s + 0.883i·43-s + 1.15i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.258 - 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.257136072\)
\(L(\frac12)\) \(\approx\) \(1.257136072\)
\(L(2)\) 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 \( 1 - 7.74T + 25T^{2} \)
7 \( 1 + 8.66T + 49T^{2} \)
11 \( 1 + 13.4T + 121T^{2} \)
13 \( 1 + 5.19iT - 169T^{2} \)
17 \( 1 - 13.4iT - 289T^{2} \)
19 \( 1 + 23iT - 361T^{2} \)
23 \( 1 - 7.74iT - 529T^{2} \)
29 \( 1 - 30.9T + 841T^{2} \)
31 \( 1 + 6.92T + 961T^{2} \)
37 \( 1 - 29.4iT - 1.36e3T^{2} \)
41 \( 1 - 80.4iT - 1.68e3T^{2} \)
43 \( 1 - 38iT - 1.84e3T^{2} \)
47 \( 1 - 54.2iT - 2.20e3T^{2} \)
53 \( 1 + 77.4T + 2.80e3T^{2} \)
59 \( 1 - 93.9T + 3.48e3T^{2} \)
61 \( 1 - 60.6iT - 3.72e3T^{2} \)
67 \( 1 - 107iT - 4.48e3T^{2} \)
71 \( 1 - 15.4iT - 5.04e3T^{2} \)
73 \( 1 + 97T + 5.32e3T^{2} \)
79 \( 1 - 67.5T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 + 174. iT - 7.92e3T^{2} \)
97 \( 1 - 109T + 9.40e3T^{2} \)
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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

−9.587026893368136201498987844806, −8.719238733588271162499448720444, −7.81789154018693854327744688919, −6.68273980784877936733999004885, −6.20023374793383929548291486979, −5.44901565927722689532919277385, −4.59038294356818743283479328561, −2.99088914252751672436658939537, −2.65293148383394911633890595707, −1.24167143392576385544126125643, 0.31776103299037399007896110072, 1.94303370916308223012812140664, 2.67480374632389948999105020585, 3.68649452024599399459345860917, 5.10243300908389727058479309004, 5.65423885717600147397184948910, 6.44961060149222597139232993006, 7.09888089491202305410171934857, 8.211583682332930630964823796808, 9.156149890887689346231239811366

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