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Properties

Label 2-60e2-20.19-c0-0-3
Degree $2$
Conductor $3600$
Sign $0.0599 + 0.998i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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  − 1.73·7-s + i·13-s − 1.73i·19-s − 1.73i·31-s − 2i·37-s + 1.73·43-s + 1.99·49-s − 61-s − 1.73·67-s − 2i·73-s − 1.73i·91-s − i·97-s − 109-s + ⋯
L(s)  = 1  − 1.73·7-s + i·13-s − 1.73i·19-s − 1.73i·31-s − 2i·37-s + 1.73·43-s + 1.99·49-s − 61-s − 1.73·67-s − 2i·73-s − 1.73i·91-s − i·97-s − 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0599 + 0.998i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.0599 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7229596349\)
\(L(\frac12)\) \(\approx\) \(0.7229596349\)
\(L(1)\) 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 \( 1 \)
good7 \( 1 + 1.73T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + 2iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−8.972812704191684147970736190894, −7.53651945173215191541791854077, −7.12128073383630226217343519377, −6.26041069058209161924812377322, −5.83001479492518291487328681494, −4.57375752149230854860579088637, −3.92124740640285264657476384108, −2.94743332839919575862384929869, −2.20882319214703421840765911854, −0.43772583544829272156730863230, 1.27362824469176788852176303353, 2.81111857194621833245509588558, 3.29369315525605633792016907440, 4.14176927804693017816704444295, 5.32054663692266614415694651963, 6.01313078680141796660719296579, 6.57247094323789106979156695380, 7.41210763121267428535238055622, 8.176836130389917512978189736210, 8.935221525037430308039466306781

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