Properties

Label 2-48e2-36.7-c0-0-2
Degree $2$
Conductor $2304$
Sign $0.766 + 0.642i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s + 17-s − 1.73i·19-s + (0.5 − 0.866i)25-s + 0.999·27-s + (1.5 − 0.866i)33-s + (−0.5 − 0.866i)41-s + (1.5 + 0.866i)43-s + (−0.5 − 0.866i)49-s + (−0.5 + 0.866i)51-s + (1.49 + 0.866i)57-s + (−1.5 + 0.866i)59-s + (1.5 − 0.866i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s + 17-s − 1.73i·19-s + (0.5 − 0.866i)25-s + 0.999·27-s + (1.5 − 0.866i)33-s + (−0.5 − 0.866i)41-s + (1.5 + 0.866i)43-s + (−0.5 − 0.866i)49-s + (−0.5 + 0.866i)51-s + (1.49 + 0.866i)57-s + (−1.5 + 0.866i)59-s + (1.5 − 0.866i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7685759548\)
\(L(\frac12)\) \(\approx\) \(0.7685759548\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.163812027834394774548878849740, −8.440198218013408817418866236911, −7.65484550460326826376738680975, −6.65135054077798821651233858875, −5.76704191023427785652996088699, −5.15989707299736134545521197121, −4.45583216891991361716697981538, −3.27240944795662776286839555987, −2.63919403612049682151273130400, −0.58801804584055334542907052762, 1.36416374581958776184885824734, 2.36554791937411813960380152206, 3.40602800127778600023832834874, 4.74848220071396319869391504358, 5.45497924284220093302993069538, 6.06769170648307973716908360533, 7.10957752643818980496318820883, 7.79211370248316861524361187863, 8.076358446923671498948303426255, 9.315443038155154327358345653219

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