Properties

Label 1-300-300.179-r0-0-0
Degree $1$
Conductor $300$
Sign $0.929 - 0.368i$
Analytic cond. $1.39319$
Root an. cond. $1.39319$
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  + 7-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (0.809 + 0.587i)23-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.809 − 0.587i)37-s + (0.809 − 0.587i)41-s + 43-s + (−0.309 − 0.951i)47-s + 49-s + (0.309 + 0.951i)53-s + (−0.809 + 0.587i)59-s + ⋯
L(s)  = 1  + 7-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (0.809 + 0.587i)23-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.809 − 0.587i)37-s + (0.809 − 0.587i)41-s + 43-s + (−0.309 − 0.951i)47-s + 49-s + (0.309 + 0.951i)53-s + (−0.809 + 0.587i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(1.39319\)
Root analytic conductor: \(1.39319\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 300,\ (0:\ ),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.352995155 - 0.2580976293i\)
\(L(\frac12)\) \(\approx\) \(1.352995155 - 0.2580976293i\)
\(L(1)\) \(\approx\) \(1.168276057 - 0.09730312630i\)
\(L(1)\) \(\approx\) \(1.168276057 - 0.09730312630i\)

Euler product

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

−25.6584895331663798378003250091, −24.260451710912809614339683819403, −23.7911338659440688672567341066, −22.87193714190321423953994368811, −21.61264490144978041449347206097, −20.99705892937011766042039510393, −20.18224595352311419561502061288, −18.96426817475242317587595099230, −18.16166630911129244018977264568, −17.33564517240204352645278871568, −16.33573107566073970297966791812, −15.18537277439827035244289822737, −14.57046578986134497283384510023, −13.35269571716185612896200885177, −12.56243210010060358190232655825, −11.21737223887458755320131556701, −10.73786374775280293337420481699, −9.34697794177499596953203571007, −8.35072671043725827900479809768, −7.45970632568573710296912682412, −6.2396513955447142806167261054, −5.01894399814120592326474705731, −4.15020283988390445100207550139, −2.592621488168522664199626395021, −1.39636499356560974169716061147, 1.1009019819076149337652368811, 2.56444320326856757897095235346, 3.810637593094758185396965129811, 5.1482208416994277744460424912, 5.88929428766669397878240326683, 7.47583983945276839299934797370, 8.14951236173647345050692984603, 9.22131235942865320029304647162, 10.59268968716898875327880547140, 11.17122670107935530908629405787, 12.31606823878678031863241490501, 13.42981948984880978156635006584, 14.21363306310705126310412139746, 15.28326637895270888624812643149, 16.12705951236040375295136200597, 17.186412861922506568751889363257, 18.20283498941642131686555833791, 18.73457769315564009958278942788, 20.05698453981790641240377912047, 21.03914719893853145368914033304, 21.356118718685106598435911373836, 22.91554405559649260897575231496, 23.37369410086465076041215525363, 24.53560560538756520339409025051, 25.15869055763125763476487231043

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