Properties

Label 2-12e3-4.3-c0-0-1
Degree $2$
Conductor $1728$
Sign $i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $0$
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 − 13-s − 1.73i·19-s − 25-s + 37-s − 1.99·49-s + 61-s − 1.73i·67-s + 73-s + 1.73i·79-s + 1.73i·91-s + 97-s − 1.73i·103-s + 2·109-s + ⋯
L(s)  = 1  − 1.73i·7-s − 13-s − 1.73i·19-s − 25-s + 37-s − 1.99·49-s + 61-s − 1.73i·67-s + 73-s + 1.73i·79-s + 1.73i·91-s + 97-s − 1.73i·103-s + 2·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9477979833\)
\(L(\frac12)\) \(\approx\) \(0.9477979833\)
\(L(1)\) 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 \( 1 + T^{2} \)
7 \( 1 + 1.73iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.615462340917795172134851838264, −8.443455406363999126001194511581, −7.49363298184073124659858536979, −7.14788637360418958420857473214, −6.26522248990561709667051410290, −4.98734926646215308428529286052, −4.39354425764661822369255022095, −3.45412206338884521975917905470, −2.26899311024382299705108684468, −0.71002701286702548198936457465, 1.88351757240761996605787485158, 2.65943885781688702367402266851, 3.80139067778985993802534900300, 4.96311573602473579778484235358, 5.72506125508415380321200066592, 6.26784205467010861434491546022, 7.50737926119342817179520179736, 8.159215719417198043085060233607, 8.933819712347643075619844316162, 9.701759286508287861707756639335

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