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Properties

Label 1-4235-4235.1972-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.463 - 0.885i$
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  + (−0.976 − 0.217i)2-s + (0.994 − 0.104i)3-s + (0.905 + 0.424i)4-s + (−0.993 − 0.113i)6-s + (−0.791 − 0.610i)8-s + (0.978 − 0.207i)9-s + (0.945 + 0.327i)12-s + (−0.389 − 0.921i)13-s + (0.640 + 0.768i)16-s + (−0.703 + 0.710i)17-s + (−0.999 − 0.00951i)18-s + (0.988 − 0.151i)19-s + (−0.371 − 0.928i)23-s + (−0.851 − 0.524i)24-s + (0.179 + 0.983i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.976 − 0.217i)2-s + (0.994 − 0.104i)3-s + (0.905 + 0.424i)4-s + (−0.993 − 0.113i)6-s + (−0.791 − 0.610i)8-s + (0.978 − 0.207i)9-s + (0.945 + 0.327i)12-s + (−0.389 − 0.921i)13-s + (0.640 + 0.768i)16-s + (−0.703 + 0.710i)17-s + (−0.999 − 0.00951i)18-s + (0.988 − 0.151i)19-s + (−0.371 − 0.928i)23-s + (−0.851 − 0.524i)24-s + (0.179 + 0.983i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.463 - 0.885i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1972, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.463 - 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.462200532 - 0.8850385269i\)
\(L(\frac12)\) \(\approx\) \(1.462200532 - 0.8850385269i\)
\(L(1)\) \(\approx\) \(1.021778257 - 0.2332146498i\)
\(L(1)\) \(\approx\) \(1.021778257 - 0.2332146498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.976 - 0.217i)T \)
3 \( 1 + (0.994 - 0.104i)T \)
13 \( 1 + (-0.389 - 0.921i)T \)
17 \( 1 + (-0.703 + 0.710i)T \)
19 \( 1 + (0.988 - 0.151i)T \)
23 \( 1 + (-0.371 - 0.928i)T \)
29 \( 1 + (0.998 + 0.0570i)T \)
31 \( 1 + (0.999 - 0.0190i)T \)
37 \( 1 + (0.730 - 0.683i)T \)
41 \( 1 + (-0.198 - 0.980i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 + (0.999 - 0.00951i)T \)
53 \( 1 + (-0.768 - 0.640i)T \)
59 \( 1 + (0.749 + 0.662i)T \)
61 \( 1 + (-0.953 - 0.299i)T \)
67 \( 1 + (-0.814 + 0.580i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (-0.962 + 0.272i)T \)
79 \( 1 + (-0.997 - 0.0760i)T \)
83 \( 1 + (0.441 + 0.897i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (0.999 + 0.0285i)T \)
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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

−18.561540507357091004673592433254, −17.92662349355814883548869451113, −17.19546125026317974396556828480, −16.3753011645073166708661402352, −15.73803706292514160423804472558, −15.38420310160196577052100819815, −14.427495379251241942904883789551, −13.919238819258995618187359422616, −13.31129803764127385758133173426, −11.9921610564875582200205580837, −11.76240639628980099573728777618, −10.70022641319086730935994205344, −9.9640027450962255011210969084, −9.39044840924390266549567009520, −8.97134645179802322627808574027, −8.07002602787520434847942297643, −7.55741667059857840198004241663, −6.87133341660487472965878161485, −6.1841302297296737705615822117, −5.02788393556531518404967380940, −4.3248323187329955131773373879, −3.18133190969177482371296668135, −2.60418887063796779310645540219, −1.78304888932178938799577890740, −0.98810882702396215421441445753, 0.6479177296043241572173580942, 1.48507676514366481479981646218, 2.54547278502218808396248325129, 2.80140676553728884849072325269, 3.81032131625432173538087207010, 4.6034350135732961794367757533, 5.85075913622040130807994260814, 6.635444479853922886370081268757, 7.389792661150695869071799540034, 7.98974313868527903244252859444, 8.576498435357613494926536699622, 9.173608507943535805417578939873, 10.07125846628138329967482046783, 10.34009811401706823311037490341, 11.27341405022372824495632709770, 12.20849204232764261263152333000, 12.685251961345710717187071936136, 13.441344534289951593134864022725, 14.31713923018948786408471812424, 14.96380393528347595348344418901, 15.74701608889337116475491067567, 16.01869267724962424030562853043, 17.124581029615024620357048302224, 17.76356592830217382217725371808, 18.22426997910535581849637108373

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