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Properties

Label 1-525-525.59-r0-0-0
Degree $1$
Conductor $525$
Sign $-0.307 - 0.951i$
Analytic cond. $2.43808$
Root an. cond. $2.43808$
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  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.809 − 0.587i)8-s + (−0.669 + 0.743i)11-s + (0.309 − 0.951i)13-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (−0.913 + 0.406i)19-s + (0.809 − 0.587i)22-s + (−0.978 − 0.207i)23-s + (−0.5 + 0.866i)26-s + (0.809 − 0.587i)29-s + (0.104 − 0.994i)31-s + (−0.5 − 0.866i)32-s + (−0.309 + 0.951i)34-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.809 − 0.587i)8-s + (−0.669 + 0.743i)11-s + (0.309 − 0.951i)13-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (−0.913 + 0.406i)19-s + (0.809 − 0.587i)22-s + (−0.978 − 0.207i)23-s + (−0.5 + 0.866i)26-s + (0.809 − 0.587i)29-s + (0.104 − 0.994i)31-s + (−0.5 − 0.866i)32-s + (−0.309 + 0.951i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.307 - 0.951i$
Analytic conductor: \(2.43808\)
Root analytic conductor: \(2.43808\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 525,\ (0:\ ),\ -0.307 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3216674656 - 0.4422317816i\)
\(L(\frac12)\) \(\approx\) \(0.3216674656 - 0.4422317816i\)
\(L(1)\) \(\approx\) \(0.5914371324 - 0.1458682298i\)
\(L(1)\) \(\approx\) \(0.5914371324 - 0.1458682298i\)

Euler product

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

−23.75038297518856847065100009918, −23.420824028755289041235587147637, −21.51414470706348994795804048001, −21.48400631411565964012890927235, −20.1358238390083980423775587976, −19.404704812262030451122265323326, −18.683659716211457135081979119939, −17.90815491537978196633775243804, −16.96868380664136359693939834425, −16.25145439121199390106957055924, −15.50264079307210236406017167429, −14.52790959210752672972553529481, −13.57512647476069058050819192009, −12.37732494543056637694396824179, −11.40530570044435625771247866441, −10.599969699193492119717226639319, −9.85635736821304371838826491376, −8.490221398447010863295060094633, −8.37497739375785888129533863103, −6.90675193716501422921990704685, −6.260334152713458328021815408614, −5.143680411861926433529473651004, −3.68412971188905003005616776991, −2.42269408125341784294242450980, −1.34210262640929597889298419121, 0.40416326615022342332065930809, 1.964295584724608248881324092671, 2.82213493016333740106045097591, 4.11006719159472851646683862627, 5.507350588667137499073072980869, 6.54586456449986923336417561664, 7.64757238424943102592674806320, 8.202293352998113292771577837511, 9.350693240579230932771114117498, 10.20329023327027442858727381433, 10.81150793733866923414579209250, 12.00470943358304390954847035603, 12.63028283179929754733045041433, 13.75209268488205622279720397245, 15.087876084471035958593703908689, 15.683480916495290868688829889932, 16.56331923566204621717249194409, 17.580827391990428015297175939851, 18.11793394529846683389910864468, 18.93772899149767171766916212537, 19.90913753046582519973235884647, 20.63200067514573456238495660547, 21.19847449955291969064351812505, 22.44036197566749832897576574717, 23.224993796116007397712824391904

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