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Properties

Label 2-3240-45.22-c0-0-0
Degree $2$
Conductor $3240$
Sign $0.0572 - 0.998i$
Analytic cond. $1.61697$
Root an. cond. $1.27160$
Motivic weight $0$
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 + (0.5 + 0.866i)11-s − i·19-s + (0.366 + 1.36i)23-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (1 + i)37-s + (−0.5 + 0.866i)41-s + (−1.36 − 0.366i)43-s + (−0.366 + 1.36i)47-s + (0.866 + 0.5i)49-s − 0.999·55-s + (−0.866 − 0.5i)59-s + (−1.36 + 0.366i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)11-s − i·19-s + (0.366 + 1.36i)23-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (1 + i)37-s + (−0.5 + 0.866i)41-s + (−1.36 − 0.366i)43-s + (−0.366 + 1.36i)47-s + (0.866 + 0.5i)49-s − 0.999·55-s + (−0.866 − 0.5i)59-s + (−1.36 + 0.366i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0572 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0572 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.0572 - 0.998i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (2377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.0572 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053736669\)
\(L(\frac12)\) \(\approx\) \(1.053736669\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - iT - T^{2} \)
97 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.154473359471798235938521563944, −8.081130574678612484007986221795, −7.51428295178064479261775509314, −6.75305110262426477727581411544, −6.28876774820868120322483359642, −5.04086150914013921256310793949, −4.39966533212834919477159111666, −3.39336745468154481034985422019, −2.69975141606268938637675253478, −1.45498677274894673639016116156, 0.67546557150675981203024639222, 1.89575732645684157178202346055, 3.22670852639926538921585103035, 3.98189691052450903541596376363, 4.75008053349567689454058351269, 5.62890187839233156463048863031, 6.32112476942855722632999851409, 7.24936844627406609799534562023, 8.077962869407071539825627935258, 8.658373309967861210365298090010

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