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Properties

Label 2-12e3-12.11-c1-0-4
Degree $2$
Conductor $1728$
Sign $-i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·7-s − 7·13-s + 8.66i·19-s + 5·25-s + 10.3i·31-s + 37-s + 10.3i·43-s + 4·49-s + 13·61-s − 12.1i·67-s − 17·73-s + 12.1i·79-s + 12.1i·91-s − 5·97-s + 19.0i·103-s + ⋯
L(s)  = 1  − 0.654i·7-s − 1.94·13-s + 1.98i·19-s + 25-s + 1.86i·31-s + 0.164·37-s + 1.58i·43-s + 0.571·49-s + 1.66·61-s − 1.48i·67-s − 1.98·73-s + 1.36i·79-s + 1.27i·91-s − 0.507·97-s + 1.87i·103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.035681942\)
\(L(\frac12)\) \(\approx\) \(1.035681942\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 7T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 8.66iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13T + 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 17T + 73T^{2} \)
79 \( 1 - 12.1iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−9.731044914578094652579680699977, −8.685689624682117387766615462127, −7.82252683610666423887560073964, −7.22423939094805309808188966149, −6.43153767971182267175224459360, −5.31833471478143608636932407450, −4.63724798252951552794818153893, −3.62114283538790970671762468404, −2.59845514440278476533583409763, −1.31261338237496169219550759733, 0.39697753974990475622521791144, 2.30130999490660312373187177893, 2.78192580433682236036944984112, 4.28410944010729513450560471737, 5.02285598360198136355258917607, 5.75716120462277241242204857502, 6.99282675466006285673500884659, 7.30467197380458481747308414317, 8.470402584093670698652149355294, 9.156964595898983359691706414769

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