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Properties

Label 2-1620-9.5-c2-0-12
Degree $2$
Conductor $1620$
Sign $0.939 + 0.342i$
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 + (−1 − 1.73i)7-s + (−11.6 + 6.70i)11-s + (−4 + 6.92i)13-s − 13.4i·17-s − 34·19-s + (34.8 + 20.1i)23-s + (2.5 + 4.33i)25-s + (34.8 − 20.1i)29-s + (−7 + 12.1i)31-s + 4.47i·35-s + 56·37-s + (−23.2 − 13.4i)41-s + (−4 − 6.92i)43-s + (34.8 − 20.1i)47-s + ⋯
L(s)  = 1  + (−0.387 − 0.223i)5-s + (−0.142 − 0.247i)7-s + (−1.05 + 0.609i)11-s + (−0.307 + 0.532i)13-s − 0.789i·17-s − 1.78·19-s + (1.51 + 0.874i)23-s + (0.100 + 0.173i)25-s + (1.20 − 0.693i)29-s + (−0.225 + 0.391i)31-s + 0.127i·35-s + 1.51·37-s + (−0.566 − 0.327i)41-s + (−0.0930 − 0.161i)43-s + (0.741 − 0.428i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(44.1418\)
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: \((2,\ 1620,\ (\ :1),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.329890746\)
\(L(\frac12)\) \(\approx\) \(1.329890746\)
\(L(2)\) 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 \( 1 + (1 + 1.73i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (11.6 - 6.70i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4 - 6.92i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 + 34T + 361T^{2} \)
23 \( 1 + (-34.8 - 20.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-34.8 + 20.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (7 - 12.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 56T + 1.36e3T^{2} \)
41 \( 1 + (23.2 + 13.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-34.8 + 20.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 40.2iT - 2.80e3T^{2} \)
59 \( 1 + (-11.6 - 6.70i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-23 - 39.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (16 - 27.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 106T + 5.32e3T^{2} \)
79 \( 1 + (-11 - 19.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-104. + 60.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 107. iT - 7.92e3T^{2} \)
97 \( 1 + (61 + 105. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.072485645343362102991206626719, −8.437807614689137123879365935285, −7.41273152627741654478443806241, −7.00708120504570146824009391426, −5.89054771481605769262349083074, −4.81308458282231487069867250282, −4.34969143926718967812324082824, −3.04472140402983026520327065737, −2.11345600169845978839940176682, −0.56933853490506079762951074847, 0.68359177696895755364435949970, 2.40015969765340655680932901815, 3.07616461151210526210794678839, 4.26978936316231712690229768115, 5.09428793963583477157473912787, 6.10418873088002801847154717678, 6.75074199215423785572931436991, 7.85850236900346995391120983920, 8.368328228196266059695812416437, 9.083408605907912955379545999947

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