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Properties

Label 1-1575-1575.166-r1-0-0
Degree $1$
Conductor $1575$
Sign $-0.476 - 0.879i$
Analytic cond. $169.257$
Root an. cond. $169.257$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)11-s + (0.978 − 0.207i)13-s + (0.309 − 0.951i)16-s + (−0.913 + 0.406i)17-s + (0.104 − 0.994i)19-s + (0.913 + 0.406i)22-s + (−0.978 − 0.207i)23-s + (0.5 + 0.866i)26-s + (−0.104 − 0.994i)29-s + (0.809 + 0.587i)31-s + 32-s + (−0.669 − 0.743i)34-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)11-s + (0.978 − 0.207i)13-s + (0.309 − 0.951i)16-s + (−0.913 + 0.406i)17-s + (0.104 − 0.994i)19-s + (0.913 + 0.406i)22-s + (−0.978 − 0.207i)23-s + (0.5 + 0.866i)26-s + (−0.104 − 0.994i)29-s + (0.809 + 0.587i)31-s + 32-s + (−0.669 − 0.743i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.476 - 0.879i$
Analytic conductor: \(169.257\)
Root analytic conductor: \(169.257\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (1:\ ),\ -0.476 - 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06697644844 - 0.1124929635i\)
\(L(\frac12)\) \(\approx\) \(0.06697644844 - 0.1124929635i\)
\(L(1)\) \(\approx\) \(0.9215382066 + 0.3958555208i\)
\(L(1)\) \(\approx\) \(0.9215382066 + 0.3958555208i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.669 + 0.743i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−20.52292996005041015121978170388, −20.0116422099979228113198598232, −19.19386420813381363342593782227, −18.39800001909639304964921130059, −17.85176213483323747147429416138, −17.02850566693758705141158373251, −15.94154118053119574231147312993, −15.24128197505724915034322834571, −14.22035510577382308951442910199, −13.847573896154352753304800381337, −12.899239736460244294861331319512, −12.156299341598118612764126718860, −11.57014504322576643932044628260, −10.71253302830858340423127809583, −10.01504377151237764725073854341, −9.15216650119440341278804880234, −8.575299237768828270228829645895, −7.417630624533289193605300710591, −6.33027505028721200563324081375, −5.65610979193393297635994201768, −4.46381602087048945258385064200, −4.00716828911550468049828371493, −3.03175165464470036088300439577, −1.92224988239176839921164539387, −1.314265858124759261528492295105, 0.023297132228211504197207847572, 1.10619296557889207310600821044, 2.587958764581399717853155668256, 3.6623424719085058320412226291, 4.26144516800869441567344485140, 5.26324907137908697712492005787, 6.26988196056839242912374183118, 6.52602989339123631548932490761, 7.68324417341013069136112690853, 8.547654722794302450536558407442, 8.93163528873563001930276592955, 9.99402911091383589934479396730, 11.055628733828557022752540658969, 11.79373071529465859610052353008, 12.74083934082557645029167114936, 13.64397263145172107292929271117, 13.87062762504680363071689522304, 14.9522134417344624237448207600, 15.67927964172028573603125704104, 16.132866568006342240045233350083, 17.10777350322684629240263168370, 17.66200052816734525871749494553, 18.37790436515573624098554778817, 19.27099550708176331846329135272, 20.00779323320884100548828409282

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