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Properties

Label 2-1620-45.7-c0-0-1
Degree $2$
Conductor $1620$
Sign $0.989 + 0.144i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
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  + (0.866 − 0.5i)5-s + (1.36 − 0.366i)7-s + (−0.5 + 0.866i)11-s + (1 + i)17-s + i·19-s + (−1.36 − 0.366i)23-s + (0.499 − 0.866i)25-s + (−0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + (0.999 − i)35-s + (−0.5 − 0.866i)41-s + (0.866 − 0.5i)49-s + (−1 + i)53-s + 0.999i·55-s + (0.866 − 0.5i)59-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)5-s + (1.36 − 0.366i)7-s + (−0.5 + 0.866i)11-s + (1 + i)17-s + i·19-s + (−1.36 − 0.366i)23-s + (0.499 − 0.866i)25-s + (−0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + (0.999 − i)35-s + (−0.5 − 0.866i)41-s + (0.866 − 0.5i)49-s + (−1 + i)53-s + 0.999i·55-s + (0.866 − 0.5i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.989 + 0.144i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.989 + 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.449844911\)
\(L(\frac12)\) \(\approx\) \(1.449844911\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
good7 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + iT^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 + (0.866 - 0.5i)T^{2} \)
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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

−9.800672306050748219293590805577, −8.648936175389204588309679621492, −7.947508062030562723630350491022, −7.46467348169378646852266473623, −6.05478299350966692998965031741, −5.54546492928011380165718243405, −4.61489944507902507613442199574, −3.85248486766315455387593720460, −2.11681672933630524361109102582, −1.57088094904265804586296582513, 1.50602115763110178260499011098, 2.53009052315635872044885688369, 3.48395694502500866598262326846, 5.03523700319793670395609716822, 5.33402855952051040018954262106, 6.27849522966770459385418135438, 7.31721953322818822929572311056, 8.002816908936407566740668174371, 8.816525928423690562944984486511, 9.609157792139974415111264311541

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