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Properties

Label 1-600-600.437-r0-0-0
Degree $1$
Conductor $600$
Sign $-0.187 + 0.982i$
Analytic cond. $2.78638$
Root an. cond. $2.78638$
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  + i·7-s + (0.309 + 0.951i)11-s + (0.951 + 0.309i)13-s + (−0.587 + 0.809i)17-s + (−0.809 − 0.587i)19-s + (−0.951 + 0.309i)23-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (−0.951 − 0.309i)37-s + (−0.309 + 0.951i)41-s − i·43-s + (0.587 + 0.809i)47-s − 49-s + (0.587 + 0.809i)53-s + (−0.309 + 0.951i)59-s + ⋯
L(s)  = 1  + i·7-s + (0.309 + 0.951i)11-s + (0.951 + 0.309i)13-s + (−0.587 + 0.809i)17-s + (−0.809 − 0.587i)19-s + (−0.951 + 0.309i)23-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + (−0.951 − 0.309i)37-s + (−0.309 + 0.951i)41-s − i·43-s + (0.587 + 0.809i)47-s − 49-s + (0.587 + 0.809i)53-s + (−0.309 + 0.951i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(2.78638\)
Root analytic conductor: \(2.78638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 600,\ (0:\ ),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7290557200 + 0.8812769774i\)
\(L(\frac12)\) \(\approx\) \(0.7290557200 + 0.8812769774i\)
\(L(1)\) \(\approx\) \(0.9581828539 + 0.3072852471i\)
\(L(1)\) \(\approx\) \(0.9581828539 + 0.3072852471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + iT \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.915111164320680717744769386546, −22.10298508364604693598747524216, −21.13687878807391540372308331500, −20.3600161935599856339761734930, −19.68265171592006566916724709851, −18.67829316626851766428628977219, −17.9241394416210582474720968640, −16.925485816564227033262686348859, −16.25896775324061970178562811494, −15.47542651962774456530037585474, −14.06177148542040248230063473256, −13.86909527009794120246196390672, −12.78818959239052757942547600227, −11.744611263708895249457304811703, −10.75386997740295662409371213857, −10.294013538405907515533954724105, −8.88568805125548186496478196110, −8.29565519188084464546849966588, −7.08816139034547225510118684874, −6.335039648539876417835548786642, −5.24804727485444474071678380218, −4.00565889095732011357736940576, −3.37580291096330682717427297615, −1.86250002384697205436391848504, −0.57707702684453480879367935242, 1.63254579827465238708876617651, 2.449220606381810762300386838297, 3.85121703128010062657814660058, 4.694781030360601808844499133987, 5.982586742510255372647381235305, 6.54225903266417124296811782013, 7.85525173513644900698156364089, 8.75648495719113916422450311262, 9.450818357032158154062028126391, 10.571265693463042388271132106936, 11.48355512161730258626601007132, 12.3361865291692376959331553701, 13.08556994618272269723464104988, 14.11542180093565522971550801322, 15.191436732505437688873230861655, 15.54246701103588393046078195925, 16.67058768034948158632057204157, 17.69205719747931726909737796242, 18.22213263670459373391331957312, 19.26352832118001978306049673915, 19.91855644123027108167693046950, 20.99641704344568920619151233024, 21.6690955203315471010287246126, 22.43185859034801083714763064560, 23.35820869152754172418261597186

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