Properties

Degree $2$
Conductor $3600$
Sign $-0.5 - 0.866i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·7-s − 13-s + 1.73i·19-s − 1.73i·31-s − 2·37-s + 1.73i·43-s − 1.99·49-s − 61-s + 1.73i·67-s + 2·73-s − 1.73i·91-s − 97-s + 109-s + ⋯
L(s)  = 1  + 1.73i·7-s − 13-s + 1.73i·19-s − 1.73i·31-s − 2·37-s + 1.73i·43-s − 1.99·49-s − 61-s + 1.73i·67-s + 2·73-s − 1.73i·91-s − 97-s + 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.5 - 0.866i$
Motivic weight: \(0\)
Character: $\chi_{3600} (3151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9009694597\)
\(L(\frac12)\) \(\approx\) \(0.9009694597\)
\(L(1)\) 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 \( 1 - 1.73iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T + T^{2} \)
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 \( 1 + 2T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - 1.73iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.980017735857843812538456317398, −8.211667345403131720149641876132, −7.70009629641350050643809782021, −6.64105846156008902713675095987, −5.83527989471391811188545608707, −5.40928291195063443796998300218, −4.48193437396371786389006378033, −3.40572694725035332119634689387, −2.49080917123977941079809505530, −1.76418639871973891381928831482, 0.49691925903654501326642723863, 1.80936154711806098453555856754, 3.04733404124779388086795473369, 3.80687696692475507210282410059, 4.79065477133777642667361615446, 5.14818961518955227816194293011, 6.62151177421089918367420348919, 7.02991515684197995930810338388, 7.51034146867927298879897266666, 8.495906941142676337876323612065

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