Properties

Label 2-1620-45.4-c1-0-13
Degree $2$
Conductor $1620$
Sign $0.114 + 0.993i$
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  + (−2.23 − 0.133i)5-s + (−3.46 + 2i)7-s + (2 + 3.46i)11-s − 4i·17-s + (−3.46 − 2i)23-s + (4.96 + 0.598i)25-s + (−3 − 5.19i)29-s + (−2 + 3.46i)31-s + (7.99 − 4i)35-s + 8i·37-s + (5 − 8.66i)41-s + (3.46 − 2i)43-s + (3.46 − 2i)47-s + (4.49 − 7.79i)49-s − 12i·53-s + ⋯
L(s)  = 1  + (−0.998 − 0.0599i)5-s + (−1.30 + 0.755i)7-s + (0.603 + 1.04i)11-s − 0.970i·17-s + (−0.722 − 0.417i)23-s + (0.992 + 0.119i)25-s + (−0.557 − 0.964i)29-s + (−0.359 + 0.622i)31-s + (1.35 − 0.676i)35-s + 1.31i·37-s + (0.780 − 1.35i)41-s + (0.528 − 0.304i)43-s + (0.505 − 0.291i)47-s + (0.642 − 1.11i)49-s − 1.64i·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.114 + 0.993i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6122902706\)
\(L(\frac12)\) \(\approx\) \(0.6122902706\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.133i)T \)
good7 \( 1 + (3.46 - 2i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (3.46 + 2i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + (-5 + 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.46 + 2i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.46 + 2i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.46 + 2i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (6.92 - 4i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.283900969931876184670734098303, −8.508233419558895188401073963779, −7.48950521996018067769783514047, −6.87642332319357844494920346585, −6.08937244075495467160401847517, −4.99614455106061863003672712690, −4.06501136867536530542475526879, −3.24248490576513164055316093288, −2.18569338351149321847685241273, −0.28659014563820725111699679185, 1.02307350972753950315083690847, 2.89237734447915947076366684341, 3.81067839923792132259943720900, 4.11101741689478040180306691906, 5.73685574268315976964981436966, 6.35119855769786542914863803311, 7.25433687421910173593734547241, 7.87132038174592058558244840418, 8.856818920665433471557141993113, 9.464128997368328034281536105912

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