Properties

Label 2-1620-45.38-c1-0-19
Degree $2$
Conductor $1620$
Sign $0.116 + 0.993i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.48 − 1.67i)5-s + (0.732 + 2.73i)7-s + (−2.44 − 1.41i)11-s + (1.09 − 4.09i)13-s + (−1.41 − 1.41i)17-s − 4i·19-s + (−7.72 − 2.07i)23-s + (−0.598 − 4.96i)25-s + (4.94 − 8.57i)29-s + (4 + 6.92i)31-s + (5.65 + 2.82i)35-s + (−3 + 3i)37-s + (−1.22 + 0.707i)41-s + (11.5 − 3.10i)47-s + (−0.866 + 0.5i)49-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)5-s + (0.276 + 1.03i)7-s + (−0.738 − 0.426i)11-s + (0.304 − 1.13i)13-s + (−0.342 − 0.342i)17-s − 0.917i·19-s + (−1.61 − 0.431i)23-s + (−0.119 − 0.992i)25-s + (0.919 − 1.59i)29-s + (0.718 + 1.24i)31-s + (0.956 + 0.478i)35-s + (−0.493 + 0.493i)37-s + (−0.191 + 0.110i)41-s + (1.69 − 0.453i)47-s + (−0.123 + 0.0714i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.620344357\)
\(L(\frac12)\) \(\approx\) \(1.620344357\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 1.67i)T \)
good7 \( 1 + (-0.732 - 2.73i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.09 + 4.09i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (7.72 + 2.07i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.94 + 8.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + (1.22 - 0.707i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-11.5 + 3.10i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.07 + 7.07i)T - 53iT^{2} \)
59 \( 1 + (1.41 + 2.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.9 + 2.92i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (-7 - 7i)T + 73iT^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.10 + 11.5i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 1.41T + 89T^{2} \)
97 \( 1 + (-1.09 - 4.09i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.030631217075662760350577090489, −8.437884864803261458017446068933, −7.978676211688991537880721225193, −6.59583320780520481091520519747, −5.75851804842946926497475128344, −5.26638090719120149575292530192, −4.37398918794355801217143555273, −2.86270285547027462086645026550, −2.18275851933252993978176173930, −0.61630672688956756628791821058, 1.51453008937336108994909454266, 2.43331521916558399397102460263, 3.76458708539468814704926496897, 4.42113655001871931624261668237, 5.65192800827658845894220903689, 6.37312388379521081659495655142, 7.23042361099159651778753093407, 7.78470146965479903874083742422, 8.843037765495094867790385547061, 9.744137819803175964166276339339

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