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Properties

Label 1-4235-4235.192-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.870 + 0.492i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.924 + 0.380i)2-s + (0.743 + 0.669i)3-s + (0.710 − 0.703i)4-s + (−0.941 − 0.336i)6-s + (−0.389 + 0.921i)8-s + (0.104 + 0.994i)9-s + (0.998 − 0.0475i)12-s + (−0.931 − 0.362i)13-s + (0.00951 − 0.999i)16-s + (−0.244 + 0.969i)17-s + (−0.475 − 0.879i)18-s + (−0.0665 + 0.997i)19-s + (0.814 − 0.580i)23-s + (−0.905 + 0.424i)24-s + (0.999 − 0.0190i)26-s + (−0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (−0.924 + 0.380i)2-s + (0.743 + 0.669i)3-s + (0.710 − 0.703i)4-s + (−0.941 − 0.336i)6-s + (−0.389 + 0.921i)8-s + (0.104 + 0.994i)9-s + (0.998 − 0.0475i)12-s + (−0.931 − 0.362i)13-s + (0.00951 − 0.999i)16-s + (−0.244 + 0.969i)17-s + (−0.475 − 0.879i)18-s + (−0.0665 + 0.997i)19-s + (0.814 − 0.580i)23-s + (−0.905 + 0.424i)24-s + (0.999 − 0.0190i)26-s + (−0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.870 + 0.492i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.870 + 0.492i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2895288849 + 1.099723276i\)
\(L(\frac12)\) \(\approx\) \(0.2895288849 + 1.099723276i\)
\(L(1)\) \(\approx\) \(0.7429280029 + 0.4278732414i\)
\(L(1)\) \(\approx\) \(0.7429280029 + 0.4278732414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.924 + 0.380i)T \)
3 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (-0.931 - 0.362i)T \)
17 \( 1 + (-0.244 + 0.969i)T \)
19 \( 1 + (-0.0665 + 0.997i)T \)
23 \( 1 + (0.814 - 0.580i)T \)
29 \( 1 + (0.985 + 0.170i)T \)
31 \( 1 + (-0.548 - 0.836i)T \)
37 \( 1 + (-0.986 + 0.161i)T \)
41 \( 1 + (0.564 + 0.825i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (0.475 - 0.879i)T \)
53 \( 1 + (0.999 - 0.00951i)T \)
59 \( 1 + (0.997 + 0.0760i)T \)
61 \( 1 + (0.991 - 0.132i)T \)
67 \( 1 + (-0.690 + 0.723i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (-0.976 - 0.217i)T \)
79 \( 1 + (0.683 - 0.730i)T \)
83 \( 1 + (0.980 + 0.198i)T \)
89 \( 1 + (-0.786 - 0.618i)T \)
97 \( 1 + (-0.996 - 0.0855i)T \)
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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

−17.92376786503191517494659403784, −17.74999333558529035843007509478, −17.01760876156670039930433311273, −16.03202003548256492293518193039, −15.5234456418991366090262585062, −14.725794518682536320002636153915, −13.90379466766444639511406538595, −13.3181997883726841067017190724, −12.40418636114706081261027943589, −12.05285106467555309865168761765, −11.23479884821180373973855271574, −10.46786088934583603593201616915, −9.56166271296158638605979862837, −9.0694909907327419852049036372, −8.60856548428634002609916430228, −7.58908784187765053422719471548, −7.08345198773652943041066148422, −6.75080197766989500828181797255, −5.488255688998495752567379702198, −4.45729098040479266933501732964, −3.48575802708870491460316325623, −2.669644387396760574986860356355, −2.28308788311947336989633190140, −1.26459878270947800792394157483, −0.41711928571677634415465399761, 1.08013658770602745500859247149, 2.1242351737301900599999724945, 2.664842244238433869626583940706, 3.65035870072374043851673871156, 4.55258595222725729946086817071, 5.351551388027754811011521526956, 6.09862504359750127092334476810, 7.08760391779046143982655691058, 7.67383238373423127881161353152, 8.46998700908690238023818627684, 8.812538670004048243291002831748, 9.77404896791498092196115549132, 10.22358099142581228535246679091, 10.73072142508057111606551768402, 11.62796044541501346464501916581, 12.528023287575714829291092895106, 13.30856788162942092797218616062, 14.38200443313637248598730080245, 14.7239938752133010214337972350, 15.22463005729810017483360224373, 16.062770429987188675114006074779, 16.594371596962964841551515747624, 17.21762521940604508702986972505, 17.89199011422960616716519870020, 18.82134146983249814314456841638

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