Properties

Label 2-54e2-3.2-c2-0-46
Degree $2$
Conductor $2916$
Sign $i$
Analytic cond. $79.4552$
Root an. cond. $8.91376$
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  − 4.39i·5-s − 3.57·7-s + 14.6i·11-s + 3.91·13-s + 6.88i·17-s − 28.2·19-s − 1.29i·23-s + 5.68·25-s − 37.9i·29-s + 54.4·31-s + 15.7i·35-s − 37.0·37-s + 41.1i·41-s − 3.54·43-s − 42.8i·47-s + ⋯
L(s)  = 1  − 0.879i·5-s − 0.510·7-s + 1.33i·11-s + 0.301·13-s + 0.405i·17-s − 1.48·19-s − 0.0563i·23-s + 0.227·25-s − 1.31i·29-s + 1.75·31-s + 0.448i·35-s − 1.00·37-s + 1.00i·41-s − 0.0824·43-s − 0.910i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $i$
Analytic conductor: \(79.4552\)
Root analytic conductor: \(8.91376\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.309881696\)
\(L(\frac12)\) \(\approx\) \(1.309881696\)
\(L(2)\) 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 \( 1 + 4.39iT - 25T^{2} \)
7 \( 1 + 3.57T + 49T^{2} \)
11 \( 1 - 14.6iT - 121T^{2} \)
13 \( 1 - 3.91T + 169T^{2} \)
17 \( 1 - 6.88iT - 289T^{2} \)
19 \( 1 + 28.2T + 361T^{2} \)
23 \( 1 + 1.29iT - 529T^{2} \)
29 \( 1 + 37.9iT - 841T^{2} \)
31 \( 1 - 54.4T + 961T^{2} \)
37 \( 1 + 37.0T + 1.36e3T^{2} \)
41 \( 1 - 41.1iT - 1.68e3T^{2} \)
43 \( 1 + 3.54T + 1.84e3T^{2} \)
47 \( 1 + 42.8iT - 2.20e3T^{2} \)
53 \( 1 - 47.8iT - 2.80e3T^{2} \)
59 \( 1 + 62.0iT - 3.48e3T^{2} \)
61 \( 1 + 11.1T + 3.72e3T^{2} \)
67 \( 1 - 111.T + 4.48e3T^{2} \)
71 \( 1 + 105. iT - 5.04e3T^{2} \)
73 \( 1 - 97.0T + 5.32e3T^{2} \)
79 \( 1 - 110.T + 6.24e3T^{2} \)
83 \( 1 + 131. iT - 6.88e3T^{2} \)
89 \( 1 - 101. iT - 7.92e3T^{2} \)
97 \( 1 + 165.T + 9.40e3T^{2} \)
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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

−8.336800839268723955646749129548, −7.87197229490183103459946311019, −6.58319557170166453664355806482, −6.40131893219104419465002243937, −5.10407336969019829005220713857, −4.53777148122197099899538783912, −3.78732600265374732547320683045, −2.50889121071162008622447749331, −1.62336711267269427287349062401, −0.35949084851012476333109155005, 0.896724616056180350296276347897, 2.36741822101282058328934994007, 3.15806405298262145028085968484, 3.80028871606183258470773285760, 4.95229498717642561203362603438, 5.89213202043205986117541949379, 6.61689676321906863983824433446, 6.96664860929045348826495040536, 8.259061511092070362513389499735, 8.576429518294882976678052306670

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