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Properties

Label 2-600-120.107-c0-0-0
Degree $2$
Conductor $600$
Sign $-0.828 - 0.559i$
Analytic cond. $0.299439$
Root an. cond. $0.547210$
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  + (−0.707 + 0.707i)2-s + (−0.258 + 0.965i)3-s − 1.00i·4-s + (−0.500 − 0.866i)6-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + 1.73i·11-s + (0.965 + 0.258i)12-s − 1.00·16-s + (−0.707 + 0.707i)17-s + (0.965 − 0.258i)18-s + i·19-s + (−1.22 − 1.22i)22-s + (−0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.258 + 0.965i)3-s − 1.00i·4-s + (−0.500 − 0.866i)6-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + 1.73i·11-s + (0.965 + 0.258i)12-s − 1.00·16-s + (−0.707 + 0.707i)17-s + (0.965 − 0.258i)18-s + i·19-s + (−1.22 − 1.22i)22-s + (−0.866 + 0.500i)24-s + (0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.828 - 0.559i$
Analytic conductor: \(0.299439\)
Root analytic conductor: \(0.547210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :0),\ -0.828 - 0.559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5187940337\)
\(L(\frac12)\) \(\approx\) \(0.5187940337\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
good7 \( 1 - iT^{2} \)
11 \( 1 - 1.73iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.77632476075205934241087032854, −10.22457150227953432869521154793, −9.514677351899863268037575824303, −8.754694289198534238840156128673, −7.75401124315300017862988597836, −6.77921109451525082756886939279, −5.83560833343820260372064453635, −4.85792538980520456859987723705, −4.00331564109542064884171016440, −2.04197648193072394133171450053, 0.794170604188252699665396914671, 2.39104588902355201003887907027, 3.34758466400627477626343574810, 4.95666639274171099652156480195, 6.27054728529259406896186130181, 7.05065728408423640474681779100, 8.114893874027463234935141114274, 8.666613076126001206075702910766, 9.575443336490759219419433541141, 10.92940785550835512540532356502

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