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Properties

Label 2-12e3-24.5-c2-0-45
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  + 6·5-s + 9.94·7-s − 19.8·11-s − 9.94i·13-s + 19.8i·17-s − 7i·19-s − 42i·23-s + 11·25-s + 24·29-s + 39.7·31-s + 59.6·35-s − 49.7i·37-s − 39.7i·41-s − 50i·43-s − 6i·47-s + ⋯
L(s)  = 1  + 1.20·5-s + 1.42·7-s − 1.80·11-s − 0.765i·13-s + 1.17i·17-s − 0.368i·19-s − 1.82i·23-s + 0.440·25-s + 0.827·29-s + 1.28·31-s + 1.70·35-s − 1.34i·37-s − 0.970i·41-s − 1.16i·43-s − 0.127i·47-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: $\chi_{1728} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.722999124\)
\(L(\frac12)\) \(\approx\) \(2.722999124\)
\(L(2)\) 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 \( 1 - 6T + 25T^{2} \)
7 \( 1 - 9.94T + 49T^{2} \)
11 \( 1 + 19.8T + 121T^{2} \)
13 \( 1 + 9.94iT - 169T^{2} \)
17 \( 1 - 19.8iT - 289T^{2} \)
19 \( 1 + 7iT - 361T^{2} \)
23 \( 1 + 42iT - 529T^{2} \)
29 \( 1 - 24T + 841T^{2} \)
31 \( 1 - 39.7T + 961T^{2} \)
37 \( 1 + 49.7iT - 1.36e3T^{2} \)
41 \( 1 + 39.7iT - 1.68e3T^{2} \)
43 \( 1 + 50iT - 1.84e3T^{2} \)
47 \( 1 + 6iT - 2.20e3T^{2} \)
53 \( 1 - 84T + 2.80e3T^{2} \)
59 \( 1 + 19.8T + 3.48e3T^{2} \)
61 \( 1 - 89.5iT - 3.72e3T^{2} \)
67 \( 1 + 53iT - 4.48e3T^{2} \)
71 \( 1 - 60iT - 5.04e3T^{2} \)
73 \( 1 - 119T + 5.32e3T^{2} \)
79 \( 1 - 49.7T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 59.6iT - 7.92e3T^{2} \)
97 \( 1 - 13T + 9.40e3T^{2} \)
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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

−8.800616683792616077879953541210, −8.300548483137489883753000301044, −7.65702252407031282406019409336, −6.53156629179187405692139496427, −5.54230557569772091489237232919, −5.19321346306289612587353957311, −4.21212185386230014981201741414, −2.56051340087789879806288589893, −2.16058178457753824547247158990, −0.74353474259295494456743735795, 1.23066145279136728589061744919, 2.14841881686981043714782784155, 2.98193096828007575204490820530, 4.67091040890458965049690082878, 5.07628855530289842139274882983, 5.82824674461551055950273507273, 6.85502841510379597487709038491, 7.892786930082086947735391091576, 8.206551070980566098597385908195, 9.460339982657150508908585239219

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