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Properties

Label 2-12e3-9.4-c1-0-15
Degree $2$
Conductor $1728$
Sign $-0.766 + 0.642i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)5-s + (−0.5 − 0.866i)7-s + (1.5 + 2.59i)11-s + (−0.5 + 0.866i)13-s − 6·17-s − 4·19-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s + (−1.5 − 2.59i)29-s + (2.5 − 4.33i)31-s + 3·35-s − 2·37-s + (1.5 − 2.59i)41-s + (0.5 + 0.866i)43-s + (4.5 + 7.79i)47-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)5-s + (−0.188 − 0.327i)7-s + (0.452 + 0.783i)11-s + (−0.138 + 0.240i)13-s − 1.45·17-s − 0.917·19-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s + (−0.278 − 0.482i)29-s + (0.449 − 0.777i)31-s + 0.507·35-s − 0.328·37-s + (0.234 − 0.405i)41-s + (0.0762 + 0.132i)43-s + (0.656 + 1.13i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (5.5 + 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.063196305387430526323735278891, −8.128155876327069990741301532635, −7.22119622266404512003136489359, −6.77881056155063989130879646186, −6.07622589697483140389319240124, −4.49802197748974167120918246042, −4.12326989797127292153977972366, −2.93757961303837670743908667639, −2.01706413999127714494742507875, 0, 1.35424299100842661621831154182, 2.76691368934957033057719317574, 3.93654456373886626854253836793, 4.60665795494598524692247744589, 5.51149937409174943651187173002, 6.40425999128391722405590490047, 7.28547003918835309848693603263, 8.351374497514730569535723608801, 8.776211789293068538929341667276

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