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Properties

Label 1-4235-4235.3009-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.687 + 0.726i$
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  + (0.974 + 0.226i)2-s + (0.309 + 0.951i)3-s + (0.897 + 0.441i)4-s + (0.0855 + 0.996i)6-s + (0.774 + 0.633i)8-s + (−0.809 + 0.587i)9-s + (−0.142 + 0.989i)12-s + (0.466 + 0.884i)13-s + (0.610 + 0.791i)16-s + (−0.198 + 0.980i)17-s + (−0.921 + 0.389i)18-s + (0.993 + 0.113i)19-s + (0.959 + 0.281i)23-s + (−0.362 + 0.931i)24-s + (0.254 + 0.967i)26-s + (−0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.974 + 0.226i)2-s + (0.309 + 0.951i)3-s + (0.897 + 0.441i)4-s + (0.0855 + 0.996i)6-s + (0.774 + 0.633i)8-s + (−0.809 + 0.587i)9-s + (−0.142 + 0.989i)12-s + (0.466 + 0.884i)13-s + (0.610 + 0.791i)16-s + (−0.198 + 0.980i)17-s + (−0.921 + 0.389i)18-s + (0.993 + 0.113i)19-s + (0.959 + 0.281i)23-s + (−0.362 + 0.931i)24-s + (0.254 + 0.967i)26-s + (−0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.687 + 0.726i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.687 + 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.678340167 + 3.900623840i\)
\(L(\frac12)\) \(\approx\) \(1.678340167 + 3.900623840i\)
\(L(1)\) \(\approx\) \(1.812089586 + 1.364408369i\)
\(L(1)\) \(\approx\) \(1.812089586 + 1.364408369i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.974 + 0.226i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.466 + 0.884i)T \)
17 \( 1 + (-0.198 + 0.980i)T \)
19 \( 1 + (0.993 + 0.113i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.736 + 0.676i)T \)
31 \( 1 + (-0.696 - 0.717i)T \)
37 \( 1 + (0.985 + 0.170i)T \)
41 \( 1 + (0.516 - 0.856i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.921 - 0.389i)T \)
53 \( 1 + (-0.610 + 0.791i)T \)
59 \( 1 + (-0.516 - 0.856i)T \)
61 \( 1 + (0.974 - 0.226i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (0.564 - 0.825i)T \)
79 \( 1 + (0.998 - 0.0570i)T \)
83 \( 1 + (-0.941 - 0.336i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.362 + 0.931i)T \)
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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

−18.097275707127871241591728942867, −17.7388049205085866374409305754, −16.57597095505540398790959884712, −15.97759235287924924496140822990, −15.2262980025819198200019393375, −14.51472690451850216896361715949, −13.96560743168969454002497269453, −13.28187828006010782330713246346, −12.825955418187720176065293712222, −12.182648079088818412830962903388, −11.3253090430716606698492073222, −11.03403789691454865732085797076, −9.86968989380166220611319565534, −9.205616510827666340435394014284, −8.13320164204559083955158813138, −7.56847173530950767611384290451, −6.83833826257392363614820818028, −6.20873677908100584850313298311, −5.42685404973369927775500570213, −4.78734654690464180360776910319, −3.70581497453080697514076790427, −2.85851810737474795884848073748, −2.617213912059260184459631076644, −1.34000949134938169496908692996, −0.81083733944579510053154157268, 1.38519023874047244129121314922, 2.31177886896822421390704556501, 3.14652602749982047695175814401, 3.81178212679858683091922249934, 4.372318471219815092546471213975, 5.17500232481141095561833378649, 5.7965729548762261450225068348, 6.59002605309186339437386531430, 7.43282810559099013020379217770, 8.20920857196716096405146704690, 8.96635581024810867831949404681, 9.65475117362179361389617907079, 10.638230516034287028852833418456, 11.12194675416383499078636795756, 11.72313630649666006174812969514, 12.64662012510443749943215043372, 13.32537080616335928667463282076, 14.07467015475260940546464684335, 14.50398886549267352413386598008, 15.21337425145938887382941098858, 15.83759539706029263879365466872, 16.35547700624698806191845454364, 16.98341323108723891790957133912, 17.66488160040112262470600289879, 18.83610339369566305188800683229

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