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Properties

Label 2-1620-45.29-c2-0-44
Degree $2$
Conductor $1620$
Sign $-0.691 + 0.722i$
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  + (2.21 + 4.48i)5-s + (8.02 + 4.63i)7-s + (−13.2 − 7.64i)11-s + (2.59 − 1.5i)13-s − 4.00·17-s − 27.7·19-s + (−18.9 − 32.7i)23-s + (−15.2 + 19.8i)25-s + (−39.7 − 22.9i)29-s + (−14.3 − 24.9i)31-s + (−3.02 + 46.2i)35-s + 34.3i·37-s + (23.6 − 13.6i)41-s + (−31.4 − 18.1i)43-s + (10.0 − 17.3i)47-s + ⋯
L(s)  = 1  + (0.442 + 0.896i)5-s + (1.14 + 0.661i)7-s + (−1.20 − 0.695i)11-s + (0.199 − 0.115i)13-s − 0.235·17-s − 1.46·19-s + (−0.823 − 1.42i)23-s + (−0.608 + 0.793i)25-s + (−1.37 − 0.791i)29-s + (−0.464 − 0.804i)31-s + (−0.0865 + 1.32i)35-s + 0.927i·37-s + (0.577 − 0.333i)41-s + (−0.730 − 0.421i)43-s + (0.213 − 0.369i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.691 + 0.722i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3290193339\)
\(L(\frac12)\) \(\approx\) \(0.3290193339\)
\(L(2)\) 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 \( 1 + (-2.21 - 4.48i)T \)
good7 \( 1 + (-8.02 - 4.63i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (13.2 + 7.64i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.59 + 1.5i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 4.00T + 289T^{2} \)
19 \( 1 + 27.7T + 361T^{2} \)
23 \( 1 + (18.9 + 32.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (39.7 + 22.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.3 + 24.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 34.3iT - 1.36e3T^{2} \)
41 \( 1 + (-23.6 + 13.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (31.4 + 18.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-10.0 + 17.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 57.9T + 2.80e3T^{2} \)
59 \( 1 + (-73.8 + 42.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-18.8 + 32.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (50.4 - 29.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 125. iT - 5.04e3T^{2} \)
73 \( 1 - 31.6iT - 5.32e3T^{2} \)
79 \( 1 + (19.5 - 33.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-32.9 + 57.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 85.2iT - 7.92e3T^{2} \)
97 \( 1 + (-140. - 80.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−8.636229140085280686034152653232, −8.254258129968808244374203327002, −7.42130886855693860639548870580, −6.28655160778200925162246129312, −5.76842415523580429546416645999, −4.86150476481988604999363510173, −3.78171288977179015281246540298, −2.44694670153675557187069048141, −2.06645268489892767033800805086, −0.07839728645365816911195624261, 1.51467918312123140623561014543, 2.12207440177185560275190455798, 3.79374021435837587387664299906, 4.67035045716508819120636076724, 5.22632042445070898435759169986, 6.12390024272169419912802265893, 7.42525519703219476599175415378, 7.82520624573822293818942009531, 8.703529246632704600685630194383, 9.418819903732377486776845879056

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