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Properties

Label 2-1620-45.14-c2-0-15
Degree $2$
Conductor $1620$
Sign $0.993 - 0.114i$
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  + (−4.99 − 0.299i)5-s + (−6.92 + 4i)7-s + (7.74 − 4.47i)11-s + (−10.3 − 6i)13-s − 31.3·17-s − 6·19-s + (−2.23 + 3.87i)23-s + (24.8 + 2.99i)25-s + (23.2 − 13.4i)29-s + (−17 + 29.4i)31-s + (35.7 − 17.8i)35-s − 44i·37-s + (15.4 + 8.94i)41-s + (−24.2 + 14i)43-s + (−2.23 − 3.87i)47-s + ⋯
L(s)  = 1  + (−0.998 − 0.0599i)5-s + (−0.989 + 0.571i)7-s + (0.704 − 0.406i)11-s + (−0.799 − 0.461i)13-s − 1.84·17-s − 0.315·19-s + (−0.0972 + 0.168i)23-s + (0.992 + 0.119i)25-s + (0.801 − 0.462i)29-s + (−0.548 + 0.949i)31-s + (1.02 − 0.511i)35-s − 1.18i·37-s + (0.377 + 0.218i)41-s + (−0.563 + 0.325i)43-s + (−0.0475 − 0.0824i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.993 - 0.114i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8217612671\)
\(L(\frac12)\) \(\approx\) \(0.8217612671\)
\(L(2)\) 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 \( 1 + (4.99 + 0.299i)T \)
good7 \( 1 + (6.92 - 4i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-7.74 + 4.47i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (10.3 + 6i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 31.3T + 289T^{2} \)
19 \( 1 + 6T + 361T^{2} \)
23 \( 1 + (2.23 - 3.87i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-23.2 + 13.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (17 - 29.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 44iT - 1.36e3T^{2} \)
41 \( 1 + (-15.4 - 8.94i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (24.2 - 14i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (2.23 + 3.87i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 40.2T + 2.80e3T^{2} \)
59 \( 1 + (-85.2 - 49.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (37 + 64.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (79.6 + 46i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 53.6iT - 5.04e3T^{2} \)
73 \( 1 - 56iT - 5.32e3T^{2} \)
79 \( 1 + (-39 - 67.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-51.4 - 89.0i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 17.8iT - 7.92e3T^{2} \)
97 \( 1 + (-27.7 + 16i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−8.984311555410906946208026471858, −8.668718726859497662008470160475, −7.57976307179455369149777545039, −6.77593041351800838003478677605, −6.16679877193876681872920722427, −4.99267856480951247035700046496, −4.11901741216515393287460392789, −3.24499585474696822617405587688, −2.29096900695283609188987385238, −0.49481356376974540261332416849, 0.46954036144229912448394498586, 2.14530454207462955793769418553, 3.30138062992587912599954109700, 4.23238308082666730392481363458, 4.68277073774508010662633935160, 6.26720694056010593501502934360, 6.89450717874214545833461101138, 7.34773353828930802397452326727, 8.525956922840036723094510334103, 9.112118281865325616384606007813

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