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Properties

Label 2-1620-45.38-c1-0-5
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

\(L(1)\) \(\approx\) \(1.482540967\)
\(L(\frac12)\) \(\approx\) \(1.482540967\)
\(L(\frac{3}{2})\) 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

−9.460562393777867265444859906944, −8.792801784124831268681017215066, −8.062901775488826428793193674420, −7.14284049747048404432694927843, −6.54209622679687783578175728161, −5.44650831039500697092667379194, −4.74378816905114197232909352868, −3.37303052535284331980961969800, −2.90139987027001436990363716637, −1.39070769953331276150063168173, 0.63809569458482079840168057041, 1.70185202177236488028992702494, 3.45535795977494386197080215221, 4.14274399444143315315131799340, 4.78971423442203951469245835169, 5.97983246283148678911589432127, 6.87053341763805285157741558272, 7.66595337685912205531075376090, 8.324502223892064996444381229520, 9.193859125807917117048408967306

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