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Properties

Label 2-1620-12.11-c1-0-14
Degree $2$
Conductor $1620$
Sign $-0.757 - 0.652i$
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  + (−1.32 − 0.492i)2-s + (1.51 + 1.30i)4-s + i·5-s + 3.84i·7-s + (−1.36 − 2.47i)8-s + (0.492 − 1.32i)10-s + 2.81·11-s − 5.79·13-s + (1.89 − 5.10i)14-s + (0.589 + 3.95i)16-s − 2.42i·17-s + 4.07i·19-s + (−1.30 + 1.51i)20-s + (−3.73 − 1.38i)22-s + 4.34·23-s + ⋯
L(s)  = 1  + (−0.937 − 0.348i)2-s + (0.757 + 0.652i)4-s + 0.447i·5-s + 1.45i·7-s + (−0.482 − 0.875i)8-s + (0.155 − 0.419i)10-s + 0.849·11-s − 1.60·13-s + (0.506 − 1.36i)14-s + (0.147 + 0.989i)16-s − 0.588i·17-s + 0.934i·19-s + (−0.292 + 0.338i)20-s + (−0.796 − 0.296i)22-s + 0.906·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.757 - 0.652i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.757 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6420492837\)
\(L(\frac12)\) \(\approx\) \(0.6420492837\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.492i)T \)
3 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 - 3.84iT - 7T^{2} \)
11 \( 1 - 2.81T + 11T^{2} \)
13 \( 1 + 5.79T + 13T^{2} \)
17 \( 1 + 2.42iT - 17T^{2} \)
19 \( 1 - 4.07iT - 19T^{2} \)
23 \( 1 - 4.34T + 23T^{2} \)
29 \( 1 - 7.01iT - 29T^{2} \)
31 \( 1 - 2.25iT - 31T^{2} \)
37 \( 1 - 4.29T + 37T^{2} \)
41 \( 1 - 0.109iT - 41T^{2} \)
43 \( 1 + 0.205iT - 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 + 7.55iT - 53T^{2} \)
59 \( 1 + 8.33T + 59T^{2} \)
61 \( 1 - 6.12T + 61T^{2} \)
67 \( 1 - 5.18iT - 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + 4.99iT - 79T^{2} \)
83 \( 1 + 7.71T + 83T^{2} \)
89 \( 1 - 0.134iT - 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−9.612871794979747475643327915948, −9.032164769406591795551676160136, −8.319500735636068975202058114610, −7.33303538278602159924437195260, −6.74781601242476536584682257231, −5.78443451384822491374671582032, −4.78322896342249350266018303751, −3.30283324988282913515208966509, −2.62431804035387222757010136793, −1.61656941971068358627778435733, 0.34178343857605469498797614532, 1.42963946752113256528867761599, 2.76114859186531470311413439952, 4.20830715289346171611917719332, 4.89884111317243762060965462477, 6.11807296439809102345790956387, 6.96046070241896911501687410653, 7.46882097421895401121858147620, 8.214968106373009058108140164337, 9.246259601606496330405587100102

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