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Properties

Label 2-1620-45.43-c0-0-1
Degree $2$
Conductor $1620$
Sign $0.313 + 0.949i$
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 + (−0.366 − 1.36i)7-s + (0.5 − 0.866i)11-s + (−1 + i)17-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 − 0.999i)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 + (−0.366 − 1.36i)7-s + (0.5 − 0.866i)11-s + (−1 + i)17-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 − 0.999i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.313 + 0.949i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1513, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.313 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.198262084\)
\(L(\frac12)\) \(\approx\) \(1.198262084\)
\(L(1)\) 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 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 - iT^{2} \)
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 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
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

−9.477696916675118765589103697713, −8.768992354585789474233688418528, −7.85041996634883544234535614353, −6.91757879496640183242624868892, −6.21749444399808995199126167357, −5.44038563481239592667121936743, −4.25095611198084798298916075074, −3.66200726305806221279222612118, −2.21253849749439331465534944653, −0.965515409643568353301506364935, 1.98630893470552478282824095994, 2.50649060940919929237898850424, 3.74351710674456036782814021658, 5.01201311290288263849517816199, 5.68634658767595129188056492002, 6.58839572943138311146875250768, 7.05266541565839569777895667752, 8.362420421001108360745131543343, 9.153999864824212294491668958772, 9.556984584527827372273327438420

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