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Properties

Label 2-1620-15.14-c2-0-24
Degree $2$
Conductor $1620$
Sign $0.647 + 0.762i$
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  + (−3.23 − 3.81i)5-s + 12.6i·7-s − 18.7i·11-s + 6.79i·13-s − 11.5·17-s + 24.2·19-s − 29.2·23-s + (−4.04 + 24.6i)25-s + 14.9i·29-s + 13.8·31-s + (48.2 − 40.9i)35-s − 38.2i·37-s − 30.9i·41-s + 13.0i·43-s − 5.68·47-s + ⋯
L(s)  = 1  + (−0.647 − 0.762i)5-s + 1.80i·7-s − 1.70i·11-s + 0.522i·13-s − 0.679·17-s + 1.27·19-s − 1.27·23-s + (−0.161 + 0.986i)25-s + 0.515i·29-s + 0.446·31-s + (1.37 − 1.17i)35-s − 1.03i·37-s − 0.754i·41-s + 0.304i·43-s − 0.121·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.647 + 0.762i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.647 + 0.762i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.363758159\)
\(L(\frac12)\) \(\approx\) \(1.363758159\)
\(L(2)\) 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 \( 1 + (3.23 + 3.81i)T \)
good7 \( 1 - 12.6iT - 49T^{2} \)
11 \( 1 + 18.7iT - 121T^{2} \)
13 \( 1 - 6.79iT - 169T^{2} \)
17 \( 1 + 11.5T + 289T^{2} \)
19 \( 1 - 24.2T + 361T^{2} \)
23 \( 1 + 29.2T + 529T^{2} \)
29 \( 1 - 14.9iT - 841T^{2} \)
31 \( 1 - 13.8T + 961T^{2} \)
37 \( 1 + 38.2iT - 1.36e3T^{2} \)
41 \( 1 + 30.9iT - 1.68e3T^{2} \)
43 \( 1 - 13.0iT - 1.84e3T^{2} \)
47 \( 1 + 5.68T + 2.20e3T^{2} \)
53 \( 1 - 57.7T + 2.80e3T^{2} \)
59 \( 1 - 30.3iT - 3.48e3T^{2} \)
61 \( 1 - 77.0T + 3.72e3T^{2} \)
67 \( 1 + 128. iT - 4.48e3T^{2} \)
71 \( 1 + 35.8iT - 5.04e3T^{2} \)
73 \( 1 - 40.6iT - 5.32e3T^{2} \)
79 \( 1 - 140.T + 6.24e3T^{2} \)
83 \( 1 - 118.T + 6.88e3T^{2} \)
89 \( 1 - 75.0iT - 7.92e3T^{2} \)
97 \( 1 + 84.5iT - 9.40e3T^{2} \)
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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

−8.985412958470577876675374466153, −8.449291432101465544966222337801, −7.79737641290422886829464738953, −6.51545489861005759848056510438, −5.65482356603385203473880841693, −5.21475133252720193326575791093, −3.97400957628056202672464020164, −3.06692781418488905252051208241, −1.95369887354664689215617635374, −0.48875562620154952563947044030, 0.863530437023249087969837704126, 2.28444868701897345790975061298, 3.54017506482145529643310236137, 4.18865275476274170077395684991, 4.93080299678636910005159562645, 6.41625446811213576290833537719, 7.06828223352265096715295650321, 7.59521692304039195431143102951, 8.173140286747641429752331486147, 9.749196507924527919170637126958

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