Properties

Label 1-4235-4235.1203-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.932 - 0.360i$
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  + (−0.389 + 0.921i)2-s + (−0.587 + 0.809i)3-s + (−0.696 − 0.717i)4-s + (−0.516 − 0.856i)6-s + (0.931 − 0.362i)8-s + (−0.309 − 0.951i)9-s + (0.989 − 0.142i)12-s + (−0.441 − 0.897i)13-s + (−0.0285 + 0.999i)16-s + (0.676 − 0.736i)17-s + (0.996 + 0.0855i)18-s + (0.198 − 0.980i)19-s + (−0.281 − 0.959i)23-s + (−0.254 + 0.967i)24-s + (0.998 − 0.0570i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.389 + 0.921i)2-s + (−0.587 + 0.809i)3-s + (−0.696 − 0.717i)4-s + (−0.516 − 0.856i)6-s + (0.931 − 0.362i)8-s + (−0.309 − 0.951i)9-s + (0.989 − 0.142i)12-s + (−0.441 − 0.897i)13-s + (−0.0285 + 0.999i)16-s + (0.676 − 0.736i)17-s + (0.996 + 0.0855i)18-s + (0.198 − 0.980i)19-s + (−0.281 − 0.959i)23-s + (−0.254 + 0.967i)24-s + (0.998 − 0.0570i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.932 - 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7267294828 - 0.1357115886i\)
\(L(\frac12)\) \(\approx\) \(0.7267294828 - 0.1357115886i\)
\(L(1)\) \(\approx\) \(0.6106076753 + 0.2498731861i\)
\(L(1)\) \(\approx\) \(0.6106076753 + 0.2498731861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.389 + 0.921i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (-0.441 - 0.897i)T \)
17 \( 1 + (0.676 - 0.736i)T \)
19 \( 1 + (0.198 - 0.980i)T \)
23 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (0.870 + 0.491i)T \)
31 \( 1 + (0.985 - 0.170i)T \)
37 \( 1 + (-0.884 + 0.466i)T \)
41 \( 1 + (-0.974 + 0.226i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (-0.996 + 0.0855i)T \)
53 \( 1 + (0.999 - 0.0285i)T \)
59 \( 1 + (0.974 + 0.226i)T \)
61 \( 1 + (0.921 - 0.389i)T \)
67 \( 1 + (0.755 + 0.654i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.791 - 0.610i)T \)
79 \( 1 + (-0.774 - 0.633i)T \)
83 \( 1 + (0.825 + 0.564i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (-0.967 - 0.254i)T \)
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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

−18.59850311524337425162461982995, −17.69608698471115308251579068430, −17.34170376944104019058875326928, −16.63699245274317557590669974867, −16.07196721493592598413607156559, −14.86487398414554387186039040324, −13.84763268141046589738842158589, −13.72684828584506438190584056255, −12.66361087332268123388778513367, −12.10284023716742881209754219963, −11.77053718776420081481743930693, −11.01090261409228381075618771226, −10.02344868962454027838259766358, −9.895620571764411812845758386091, −8.501349233437138668331674784559, −8.246223194589580955410743800667, −7.2982651817927848700675706453, −6.70313810539326920016444474747, −5.66604148191567221774212023097, −5.039765519529808312090031940085, −4.05654775278283091887913570226, −3.34303135328319716775941414441, −2.24038738176419321901856626872, −1.715845154461543432688691151695, −0.93152533364864347811371412262, 0.34956559167172718896850312667, 1.10413867389804073417393986800, 2.65826120469134044317971949847, 3.46968196500859127213108098698, 4.5828216245093786228013704611, 5.00739108576729219309774829015, 5.57455367206919806957724824871, 6.58099130337283181165952968429, 6.92561850468630580269521158353, 8.06381009205636730304775004453, 8.553613781864671808076783537959, 9.444906485176391778464751386643, 10.11426608442249403981873663774, 10.39196960510351197212127846194, 11.42482207136208696113964464623, 12.08355978244902352978740464610, 12.99338393898820199060460237233, 13.785259330627215066170452817104, 14.61619564480862633275265264353, 15.03283256756848640055435476019, 15.86197519206405177688555849212, 16.174773541154897206552169291748, 16.96247708121380473264103307606, 17.56015120289410255062995610430, 18.027126584725320222601033221825

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