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Properties

Label 2-1620-9.2-c2-0-1
Degree $2$
Conductor $1620$
Sign $-0.996 + 0.0871i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + (1.93 − 1.11i)5-s + (−6.81 + 11.7i)7-s + (−0.693 − 0.400i)11-s + (9.67 + 16.7i)13-s − 29.8i·17-s − 4.17·19-s + (−31.9 + 18.4i)23-s + (2.5 − 4.33i)25-s + (23.2 + 13.4i)29-s + (22.0 + 38.2i)31-s + 30.4i·35-s − 29.4·37-s + (−34.2 + 19.7i)41-s + (−3.31 + 5.73i)43-s + (−2.21 − 1.27i)47-s + ⋯
L(s)  = 1  + (0.387 − 0.223i)5-s + (−0.973 + 1.68i)7-s + (−0.0630 − 0.0363i)11-s + (0.744 + 1.28i)13-s − 1.75i·17-s − 0.219·19-s + (−1.38 + 0.800i)23-s + (0.100 − 0.173i)25-s + (0.801 + 0.462i)29-s + (0.711 + 1.23i)31-s + 0.870i·35-s − 0.796·37-s + (−0.834 + 0.481i)41-s + (−0.0770 + 0.133i)43-s + (−0.0470 − 0.0271i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.996 + 0.0871i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.996 + 0.0871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5510468332\)
\(L(\frac12)\) \(\approx\) \(0.5510468332\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + (-1.93 + 1.11i)T \)
good7 \( 1 + (6.81 - 11.7i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (0.693 + 0.400i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.67 - 16.7i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 29.8iT - 289T^{2} \)
19 \( 1 + 4.17T + 361T^{2} \)
23 \( 1 + (31.9 - 18.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-23.2 - 13.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-22.0 - 38.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 29.4T + 1.36e3T^{2} \)
41 \( 1 + (34.2 - 19.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (3.31 - 5.73i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (2.21 + 1.27i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 95.6iT - 2.80e3T^{2} \)
59 \( 1 + (-60.1 + 34.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-25.0 + 43.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (30.2 + 52.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 27.4iT - 5.04e3T^{2} \)
73 \( 1 + 106.T + 5.32e3T^{2} \)
79 \( 1 + (61.3 - 106. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (125. + 72.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 110. iT - 7.92e3T^{2} \)
97 \( 1 + (1.06 - 1.84i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.651055695530002070273248885947, −8.775439776979424480967624293601, −8.429102958139534568943449458132, −6.89178048295423032024623551843, −6.45902741818258273812211158975, −5.54090237437656029898218857344, −4.86955927349523372693237524603, −3.51614714173416993077383901404, −2.63700920676648793083330407384, −1.67259971933217873902854824266, 0.14762928990312559012543266630, 1.29742321178137240650352609207, 2.76710552134852027199974802020, 3.80265783155594422195095661943, 4.27628229104357938897486631874, 5.92971629405686059376249065412, 6.19595020548084608433072688859, 7.17739581532546530780034045897, 8.006725143720627659770703910323, 8.664175310282767469468992114238

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