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Properties

Label 2-1620-9.4-c1-0-10
Degree $2$
Conductor $1620$
Sign $0.173 + 0.984i$
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  + (0.5 − 0.866i)5-s + (−1 − 1.73i)7-s + (−1 + 1.73i)13-s + 3·17-s + 5·19-s + (1.5 − 2.59i)23-s + (−0.499 − 0.866i)25-s + (−3 − 5.19i)29-s + (−2.5 + 4.33i)31-s − 1.99·35-s + 2·37-s + (6 − 10.3i)41-s + (−4 − 6.92i)43-s + (−6 − 10.3i)47-s + (1.50 − 2.59i)49-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.377 − 0.654i)7-s + (−0.277 + 0.480i)13-s + 0.727·17-s + 1.14·19-s + (0.312 − 0.541i)23-s + (−0.0999 − 0.173i)25-s + (−0.557 − 0.964i)29-s + (−0.449 + 0.777i)31-s − 0.338·35-s + 0.328·37-s + (0.937 − 1.62i)41-s + (−0.609 − 1.05i)43-s + (−0.875 − 1.51i)47-s + (0.214 − 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.524374499\)
\(L(\frac12)\) \(\approx\) \(1.524374499\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-6 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.5 + 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.229878683424296962663505641322, −8.540608620559694123648541416883, −7.42353419162335916711038543214, −7.02402798258473775936452474587, −5.86131873523810069205181729905, −5.15833923064090687449860109703, −4.11685894741917049189733146477, −3.28028393323370994930025626178, −1.96884560660324572121077180159, −0.63043147536621173449072760429, 1.34170110184150463463131333010, 2.78976220853864652860273553189, 3.31939411859317928661593182603, 4.70216987705528096228663152106, 5.64784628382244551014933536032, 6.15296500148472383129224226925, 7.34593049514981749829029524624, 7.79115672846270028165862531331, 8.942810603845708943826866275967, 9.627096946625867461752248566673

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