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Properties

Label 2-54e2-9.4-c1-0-11
Degree $2$
Conductor $2916$
Sign $0.5 - 0.866i$
Analytic cond. $23.2843$
Root an. cond. $4.82538$
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  + (1.68 − 2.91i)5-s + (0.961 + 1.66i)7-s + (1.01 + 1.75i)11-s + (−2.94 + 5.09i)13-s + 3.39·17-s + 0.160·19-s + (−2.14 + 3.72i)23-s + (−3.15 − 5.46i)25-s + (4.14 + 7.18i)29-s + (−5.27 + 9.13i)31-s + 6.46·35-s − 4.35·37-s + (−0.644 + 1.11i)41-s + (3.61 + 6.25i)43-s + (0.0485 + 0.0840i)47-s + ⋯
L(s)  = 1  + (0.751 − 1.30i)5-s + (0.363 + 0.629i)7-s + (0.305 + 0.529i)11-s + (−0.816 + 1.41i)13-s + 0.823·17-s + 0.0368·19-s + (−0.448 + 0.776i)23-s + (−0.630 − 1.09i)25-s + (0.770 + 1.33i)29-s + (−0.947 + 1.64i)31-s + 1.09·35-s − 0.716·37-s + (−0.100 + 0.174i)41-s + (0.550 + 0.953i)43-s + (0.00707 + 0.0122i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(23.2843\)
Root analytic conductor: \(4.82538\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2916} (973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2916,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.888034150\)
\(L(\frac12)\) \(\approx\) \(1.888034150\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 2.91i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.961 - 1.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.01 - 1.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.94 - 5.09i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.39T + 17T^{2} \)
19 \( 1 - 0.160T + 19T^{2} \)
23 \( 1 + (2.14 - 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.14 - 7.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.27 - 9.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.35T + 37T^{2} \)
41 \( 1 + (0.644 - 1.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.61 - 6.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0485 - 0.0840i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + (1.21 - 2.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.88 + 4.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.07 + 3.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.19T + 71T^{2} \)
73 \( 1 + 4.25T + 73T^{2} \)
79 \( 1 + (-1.49 - 2.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.76 - 8.25i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.64T + 89T^{2} \)
97 \( 1 + (1.39 + 2.41i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.029720700210953155547637278244, −8.342185604078609950503285050165, −7.36837140030930852636215475836, −6.60707335097993683135486629886, −5.61058359724478672109802554799, −5.00416276941441560342759128375, −4.50580312620237823215921166660, −3.22713177918475375075337458961, −1.83830006989230140052673363636, −1.48142704440398285843650212646, 0.58170567922907997962137699418, 2.09405807563146730928228132728, 2.89598902352261575666258774267, 3.69040672811925790362892559750, 4.77186640809062932533108779713, 5.90627957640931785783189751126, 6.10987919777018855391951937957, 7.36429548906385789449012993140, 7.59314621315844927669717379231, 8.537688857848587865597836629635

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