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Properties

Label 2-1620-9.4-c3-0-2
Degree $2$
Conductor $1620$
Sign $-0.939 - 0.342i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.5 + 4.33i)5-s + (−1 − 1.73i)7-s + (−15 − 25.9i)11-s + (2 − 3.46i)13-s + 90·17-s − 28·19-s + (−60 + 103. i)23-s + (−12.5 − 21.6i)25-s + (−105 − 181. i)29-s + (2 − 3.46i)31-s + 10·35-s + 200·37-s + (−120 + 207. i)41-s + (68 + 117. i)43-s + (60 + 103. i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.0539 − 0.0935i)7-s + (−0.411 − 0.712i)11-s + (0.0426 − 0.0739i)13-s + 1.28·17-s − 0.338·19-s + (−0.543 + 0.942i)23-s + (−0.100 − 0.173i)25-s + (−0.672 − 1.16i)29-s + (0.0115 − 0.0200i)31-s + 0.0482·35-s + 0.888·37-s + (−0.457 + 0.791i)41-s + (0.241 + 0.417i)43-s + (0.186 + 0.322i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3331622265\)
\(L(\frac12)\) \(\approx\) \(0.3331622265\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 4.33i)T \)
good7 \( 1 + (1 + 1.73i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (15 + 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 90T + 4.91e3T^{2} \)
19 \( 1 + 28T + 6.85e3T^{2} \)
23 \( 1 + (60 - 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (105 + 181. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 200T + 5.06e4T^{2} \)
41 \( 1 + (120 - 207. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-68 - 117. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-60 - 103. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 30T + 1.48e5T^{2} \)
59 \( 1 + (-225 + 389. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-83 - 143. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (454 - 786. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + 250T + 3.89e5T^{2} \)
79 \( 1 + (-458 - 793. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-570 - 987. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 420T + 7.04e5T^{2} \)
97 \( 1 + (769 + 1.33e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.577975142715556788500628192157, −8.368239522967900127809999200971, −7.86927649750114272249377465992, −7.09163031409123613086511297252, −5.99225580246452438874083777557, −5.52172674261808539951620271367, −4.24115095677107864646307847472, −3.42904024109421979011817073283, −2.53295497382072620116038362480, −1.17473545460130699301810812103, 0.07681495809115050831755858804, 1.37232588779270444710161745246, 2.48946374736915228736623506539, 3.60785198718358548573447097410, 4.52610248911436309000694764837, 5.34890602703545337268806780692, 6.17528960484236701451022299210, 7.25866805029369186443985096867, 7.80947908342359151375510984630, 8.713671906088203592311723777997

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