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Properties

Label 2-60e2-5.4-c1-0-8
Degree $2$
Conductor $3600$
Sign $-0.447 - 0.894i$
Analytic cond. $28.7461$
Root an. cond. $5.36154$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·7-s − 2·11-s + 4i·13-s + 2i·17-s + 4·19-s − 8i·23-s + 10·29-s − 4·31-s + 8i·43-s + 8i·47-s + 3·49-s + 6i·53-s − 14·59-s − 14·61-s − 4i·67-s + ⋯
L(s)  = 1  + 0.755i·7-s − 0.603·11-s + 1.10i·13-s + 0.485i·17-s + 0.917·19-s − 1.66i·23-s + 1.85·29-s − 0.718·31-s + 1.21i·43-s + 1.16i·47-s + 0.428·49-s + 0.824i·53-s − 1.82·59-s − 1.79·61-s − 0.488i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.282756005\)
\(L(\frac12)\) \(\approx\) \(1.282756005\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 14T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−8.898732153946439668830172844087, −8.071419278667950609960640534826, −7.39092763016729078336185662304, −6.36768618406026628385527947342, −6.00342587945004892087351264539, −4.82982394260034450129292723600, −4.42668724649520779129530534448, −3.09399639507843032183673757132, −2.45575080787710399704575088270, −1.29932633844736507545975532665, 0.39613734888439158261932611325, 1.52699205641854973926152112682, 2.92055364907656699167024110686, 3.44163812686520828126498239378, 4.55739345420694758776725920803, 5.32339103551313654915036968691, 5.91980903394925545749973062305, 7.14035868448745779618555138150, 7.45137101760860090510812710529, 8.195815977761557555671030445369

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