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Properties

Label 2-1620-180.79-c0-0-8
Degree $2$
Conductor $1620$
Sign $0.939 - 0.342i$
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)2-s + (0.499 + 0.866i)4-s − i·5-s + 0.999i·8-s + (0.5 − 0.866i)10-s + (1.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s − i·17-s + (0.866 − 0.499i)20-s − 25-s + 1.73·26-s + (−0.866 + 1.5i)29-s + (−0.866 + 0.499i)32-s + (0.5 − 0.866i)34-s + 1.73i·37-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − i·5-s + 0.999i·8-s + (0.5 − 0.866i)10-s + (1.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s − i·17-s + (0.866 − 0.499i)20-s − 25-s + 1.73·26-s + (−0.866 + 1.5i)29-s + (−0.866 + 0.499i)32-s + (0.5 − 0.866i)34-s + 1.73i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.903876632\)
\(L(\frac12)\) \(\approx\) \(1.903876632\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.387412394885611001390898614251, −8.608444946544878481755644912014, −8.070232178690491686155453456184, −7.16262094035396970355334165556, −6.22031393266451291824868806856, −5.43376046133416910566317264553, −4.84190186407333219080623164658, −3.80705727403318429549802322263, −2.99898421438659897999071535336, −1.44137647800351894646726898792, 1.64338055233249409097346739135, 2.59586186248190910979691187596, 3.88156644436308382415282703005, 4.02082850120711028131731895036, 5.65183261391244173302746590622, 6.13121024246363087341627813574, 6.87417665803887921583664907984, 7.78597927458104807636715313656, 8.914939898645643276668562489691, 9.750539564565347596011197610112

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