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Properties

Label 2-1620-9.5-c2-0-31
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: $\chi_{1620} (701, \cdot )$
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

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

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