Properties

Label 2-18e2-108.103-c2-0-7
Degree $2$
Conductor $324$
Sign $0.540 - 0.841i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
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  + (0.831 − 1.81i)2-s + (−2.61 − 3.02i)4-s + (−3.67 + 1.33i)5-s + (−2.42 + 0.428i)7-s + (−7.67 + 2.24i)8-s + (−0.622 + 7.78i)10-s + (−6.11 + 16.8i)11-s + (15.7 + 13.2i)13-s + (−1.24 + 4.77i)14-s + (−2.30 + 15.8i)16-s + (11.4 − 19.8i)17-s + (−6.65 + 3.83i)19-s + (13.6 + 7.60i)20-s + (25.4 + 25.1i)22-s + (1.94 + 0.343i)23-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−0.654 − 0.756i)4-s + (−0.734 + 0.267i)5-s + (−0.346 + 0.0611i)7-s + (−0.959 + 0.280i)8-s + (−0.0622 + 0.778i)10-s + (−0.556 + 1.52i)11-s + (1.21 + 1.01i)13-s + (−0.0885 + 0.340i)14-s + (−0.143 + 0.989i)16-s + (0.674 − 1.16i)17-s + (−0.350 + 0.202i)19-s + (0.682 + 0.380i)20-s + (1.15 + 1.14i)22-s + (0.0846 + 0.0149i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.540 - 0.841i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.540 - 0.841i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.757186 + 0.413735i\)
\(L(\frac12)\) \(\approx\) \(0.757186 + 0.413735i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 + 1.81i)T \)
3 \( 1 \)
good5 \( 1 + (3.67 - 1.33i)T + (19.1 - 16.0i)T^{2} \)
7 \( 1 + (2.42 - 0.428i)T + (46.0 - 16.7i)T^{2} \)
11 \( 1 + (6.11 - 16.8i)T + (-92.6 - 77.7i)T^{2} \)
13 \( 1 + (-15.7 - 13.2i)T + (29.3 + 166. i)T^{2} \)
17 \( 1 + (-11.4 + 19.8i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (6.65 - 3.83i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-1.94 - 0.343i)T + (497. + 180. i)T^{2} \)
29 \( 1 + (33.6 - 28.2i)T + (146. - 828. i)T^{2} \)
31 \( 1 + (20.3 + 3.58i)T + (903. + 328. i)T^{2} \)
37 \( 1 + (15.3 - 26.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-4.57 - 3.83i)T + (291. + 1.65e3i)T^{2} \)
43 \( 1 + (10.9 - 29.9i)T + (-1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (-53.8 + 9.49i)T + (2.07e3 - 755. i)T^{2} \)
53 \( 1 + 19.4T + 2.80e3T^{2} \)
59 \( 1 + (19.2 + 52.7i)T + (-2.66e3 + 2.23e3i)T^{2} \)
61 \( 1 + (2.78 + 15.7i)T + (-3.49e3 + 1.27e3i)T^{2} \)
67 \( 1 + (1.41 - 1.68i)T + (-779. - 4.42e3i)T^{2} \)
71 \( 1 + (86.0 + 49.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-31.8 - 55.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-12.9 - 15.4i)T + (-1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (-33.4 - 39.9i)T + (-1.19e3 + 6.78e3i)T^{2} \)
89 \( 1 + (-8.30 - 14.3i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (120. + 43.6i)T + (7.20e3 + 6.04e3i)T^{2} \)
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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

−11.54540410481397285618925044013, −10.84942935875681409297326083051, −9.767427397292875960164825038749, −9.086895709134548632107444284032, −7.70447143596290540695310608151, −6.67244171391699384372448779781, −5.28810680335943398226932671984, −4.20968661050121950433240244364, −3.23587571075876725646988807799, −1.74556850536256845341544705360, 0.35852969316470240944876208903, 3.31610141897913432168260406981, 3.96445552067677531844130561822, 5.63565549520728414689174936144, 6.04270984746710236087835173831, 7.57177747232167282421533720120, 8.251412301436375008721520778753, 8.886723660155422123897063276914, 10.42385389214785601116723368648, 11.29939977693641625989296306542

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