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Properties

Label 2-1620-9.2-c2-0-23
Degree $2$
Conductor $1620$
Sign $0.0871 + 0.996i$
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 + (−3.54 + 6.14i)7-s + (−2.92 − 1.68i)11-s + (−1.83 − 3.17i)13-s + 7.23i·17-s − 16.2·19-s + (19.2 − 11.1i)23-s + (2.5 − 4.33i)25-s + (−17.3 − 10.0i)29-s + (−16.5 − 28.6i)31-s + 15.8i·35-s + 19.0·37-s + (27.9 − 16.1i)41-s + (10.0 − 17.4i)43-s + (3.71 + 2.14i)47-s + ⋯
L(s)  = 1  + (0.387 − 0.223i)5-s + (−0.506 + 0.877i)7-s + (−0.265 − 0.153i)11-s + (−0.141 − 0.244i)13-s + 0.425i·17-s − 0.852·19-s + (0.835 − 0.482i)23-s + (0.100 − 0.173i)25-s + (−0.599 − 0.346i)29-s + (−0.533 − 0.924i)31-s + 0.453i·35-s + 0.514·37-s + (0.681 − 0.393i)41-s + (0.234 − 0.406i)43-s + (0.0790 + 0.0456i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.0871 + 0.996i$
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: \((2,\ 1620,\ (\ :1),\ 0.0871 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.250007973\)
\(L(\frac12)\) \(\approx\) \(1.250007973\)
\(L(2)\) 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 \( 1 + (3.54 - 6.14i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (2.92 + 1.68i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (1.83 + 3.17i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 7.23iT - 289T^{2} \)
19 \( 1 + 16.2T + 361T^{2} \)
23 \( 1 + (-19.2 + 11.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (17.3 + 10.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (16.5 + 28.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 19.0T + 1.36e3T^{2} \)
41 \( 1 + (-27.9 + 16.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-10.0 + 17.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-3.71 - 2.14i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 41.4iT - 2.80e3T^{2} \)
59 \( 1 + (26.3 - 15.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-28.9 + 50.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (39.6 + 68.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 16.1iT - 5.04e3T^{2} \)
73 \( 1 - 68.9T + 5.32e3T^{2} \)
79 \( 1 + (-61.3 + 106. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (83.8 + 48.4i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 168. iT - 7.92e3T^{2} \)
97 \( 1 + (-65.3 + 113. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.086229790364541858323153509108, −8.352481886376836558966066513475, −7.47742827117202212001625566564, −6.38260036914117778204275064290, −5.84487053454917265937216272431, −5.00341131769429814448562385264, −3.92850723416178031426877582411, −2.78240457536405599968409824159, −1.97712238939126378705964955482, −0.36054073492015952366009682069, 1.09880390942233022013353408349, 2.40858518340549748004851181733, 3.41146543950705150996323496278, 4.35814584724154535209506404483, 5.29649141244018382139187625874, 6.27144133045722770850079462996, 7.05603811969022327239936954964, 7.57481051459226168765570786639, 8.749887739277109074002688316955, 9.437104979983383388087919362410

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