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Properties

Label 1-4235-4235.773-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.944 - 0.329i$
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  + (0.986 − 0.161i)2-s + (0.406 − 0.913i)3-s + (0.948 − 0.318i)4-s + (0.254 − 0.967i)6-s + (0.884 − 0.466i)8-s + (−0.669 − 0.743i)9-s + (0.0950 − 0.995i)12-s + (−0.113 − 0.993i)13-s + (0.797 − 0.603i)16-s + (−0.901 − 0.432i)17-s + (−0.780 − 0.625i)18-s + (−0.179 + 0.983i)19-s + (0.945 − 0.327i)23-s + (−0.0665 − 0.997i)24-s + (−0.272 − 0.962i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.986 − 0.161i)2-s + (0.406 − 0.913i)3-s + (0.948 − 0.318i)4-s + (0.254 − 0.967i)6-s + (0.884 − 0.466i)8-s + (−0.669 − 0.743i)9-s + (0.0950 − 0.995i)12-s + (−0.113 − 0.993i)13-s + (0.797 − 0.603i)16-s + (−0.901 − 0.432i)17-s + (−0.780 − 0.625i)18-s + (−0.179 + 0.983i)19-s + (0.945 − 0.327i)23-s + (−0.0665 − 0.997i)24-s + (−0.272 − 0.962i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5480548957 - 3.235017410i\)
\(L(\frac12)\) \(\approx\) \(0.5480548957 - 3.235017410i\)
\(L(1)\) \(\approx\) \(1.636676301 - 1.194399618i\)
\(L(1)\) \(\approx\) \(1.636676301 - 1.194399618i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.986 - 0.161i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (-0.113 - 0.993i)T \)
17 \( 1 + (-0.901 - 0.432i)T \)
19 \( 1 + (-0.179 + 0.983i)T \)
23 \( 1 + (0.945 - 0.327i)T \)
29 \( 1 + (-0.610 - 0.791i)T \)
31 \( 1 + (0.217 - 0.976i)T \)
37 \( 1 + (-0.999 - 0.00951i)T \)
41 \( 1 + (0.998 - 0.0570i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (0.780 - 0.625i)T \)
53 \( 1 + (-0.603 + 0.797i)T \)
59 \( 1 + (0.449 + 0.893i)T \)
61 \( 1 + (0.935 - 0.353i)T \)
67 \( 1 + (-0.998 - 0.0475i)T \)
71 \( 1 + (-0.0285 - 0.999i)T \)
73 \( 1 + (-0.730 + 0.683i)T \)
79 \( 1 + (-0.640 + 0.768i)T \)
83 \( 1 + (-0.856 - 0.516i)T \)
89 \( 1 + (0.235 - 0.971i)T \)
97 \( 1 + (-0.441 + 0.897i)T \)
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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

−19.08362840530097428022097887553, −17.6352363382985445360378542726, −17.187112822523536651447987417076, −16.30116372095418037081872908161, −15.903225674104433654984809440066, −15.195542847142564597848414041577, −14.638040827853643268373517903807, −14.05266054569785871262493904991, −13.33950145035347866262424426869, −12.77912602628209849437946215903, −11.77193458888207296357806274977, −11.108239741766806766312487375, −10.73901919231812884785189068319, −9.748510499653840313827938835155, −8.889813723541406468557683511875, −8.47945604809768538604068073578, −7.26102572687254543777472649329, −6.83714868987809797988889602378, −5.89467615262598074778917467561, −4.97894324707928302534102704813, −4.628284650307979139358949102978, −3.81858154147369318998246687471, −3.132263769838621263403408945177, −2.361791959274571848597261632971, −1.56409317918696082338401860788, 0.5161465074677232054403720764, 1.53892572101115401854184827248, 2.33813079008709108160984978544, 2.92850893956616030634496968519, 3.73503605071434724471921903914, 4.542100833032967430083290230014, 5.576735904915838295188581732567, 5.99948209405241791264865450959, 6.90142776264535848747746567366, 7.43962115971870534256656872978, 8.15796472791205034141019903407, 8.98377792558111189085523441072, 9.9535992570452733648681164252, 10.71488158120705923059284698308, 11.50113171093849718918222471885, 12.06459667627990322453528951686, 12.87689875274384422567178109447, 13.17977802195720383988076033630, 13.88029512851497184366473941892, 14.59632522975247837483279985293, 15.18140167312721349631311004704, 15.71005052242191576702450295466, 16.80758624142043115100156233407, 17.31120878542888523445992940586, 18.24978673878779136593412148138

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