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

\(L(\frac{1}{2})\) \(\approx\) \(1.304541734\)
\(L(\frac12)\) \(\approx\) \(1.304541734\)
\(L(1)\) 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

−8.446156141950173316095183383653, −7.74150735439921101569748939819, −7.34431156178424980269602863901, −6.25529837166572875014060175006, −5.85717804278547754481236016725, −4.53293702398748791021384047300, −3.96765265705382171201653111116, −3.35147616920509415702730398561, −1.88976216955315927353710890275, −0.849287628200568591356291987110, 1.36858935655896603797884666969, 2.61659039085290545781143792099, 3.06362229102243986605066833457, 4.41151213172758678057851222304, 5.10156698455657904980448646174, 5.96532382474648780660299390712, 6.41932832409070900917269557885, 7.39502693616187491454632631477, 8.352937788875759211196280498756, 8.867470330904173342821232056490

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