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Properties

Label 1-4235-4235.2624-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.478 - 0.878i$
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  + (0.0855 − 0.996i)2-s + (0.309 + 0.951i)3-s + (−0.985 − 0.170i)4-s + (0.974 − 0.226i)6-s + (−0.254 + 0.967i)8-s + (−0.809 + 0.587i)9-s + (−0.142 − 0.989i)12-s + (−0.696 − 0.717i)13-s + (0.941 + 0.336i)16-s + (0.870 − 0.491i)17-s + (0.516 + 0.856i)18-s + (−0.736 + 0.676i)19-s + (0.959 − 0.281i)23-s + (−0.998 + 0.0570i)24-s + (−0.774 + 0.633i)26-s + (−0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.0855 − 0.996i)2-s + (0.309 + 0.951i)3-s + (−0.985 − 0.170i)4-s + (0.974 − 0.226i)6-s + (−0.254 + 0.967i)8-s + (−0.809 + 0.587i)9-s + (−0.142 − 0.989i)12-s + (−0.696 − 0.717i)13-s + (0.941 + 0.336i)16-s + (0.870 − 0.491i)17-s + (0.516 + 0.856i)18-s + (−0.736 + 0.676i)19-s + (0.959 − 0.281i)23-s + (−0.998 + 0.0570i)24-s + (−0.774 + 0.633i)26-s + (−0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.478 - 0.878i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2624, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.478 - 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.157445442 - 0.6877602798i\)
\(L(\frac12)\) \(\approx\) \(1.157445442 - 0.6877602798i\)
\(L(1)\) \(\approx\) \(0.9677607264 - 0.2320738808i\)
\(L(1)\) \(\approx\) \(0.9677607264 - 0.2320738808i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.0855 - 0.996i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.696 - 0.717i)T \)
17 \( 1 + (0.870 - 0.491i)T \)
19 \( 1 + (-0.736 + 0.676i)T \)
23 \( 1 + (0.959 - 0.281i)T \)
29 \( 1 + (-0.993 + 0.113i)T \)
31 \( 1 + (0.466 + 0.884i)T \)
37 \( 1 + (-0.897 - 0.441i)T \)
41 \( 1 + (-0.921 - 0.389i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (0.516 - 0.856i)T \)
53 \( 1 + (-0.941 + 0.336i)T \)
59 \( 1 + (0.921 - 0.389i)T \)
61 \( 1 + (0.0855 + 0.996i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (0.198 - 0.980i)T \)
73 \( 1 + (0.0285 - 0.999i)T \)
79 \( 1 + (0.362 - 0.931i)T \)
83 \( 1 + (-0.610 - 0.791i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (-0.998 + 0.0570i)T \)
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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

−18.58519225289551259039869666083, −17.593883898411579895092657867303, −17.00969757231700317752918464105, −16.85304726241248292666026546721, −15.545999438450371451981500931481, −15.12165141790478262927466578096, −14.3022599574056583665742089691, −13.99131144956554598151682597099, −12.919999310747480186475852658395, −12.81584802339734436958657806378, −11.85018679751051986448988233638, −11.11999163529319813455095401451, −9.89753386663268651872284929305, −9.33974080309279926282833665501, −8.54009544178748699203929594243, −8.04094792779096381633057764594, −7.06512130212157878809302305416, −6.95076318131776607267260554435, −5.95555836655372609792219883967, −5.35224053432350776570850290122, −4.42387177449355777584446406296, −3.59756316137450533842900312723, −2.70750246049848887817577982857, −1.745634139048780018441140483888, −0.740548546656900328988883128035, 0.48063485932422739579355819577, 1.73023814299616553471707634604, 2.57448603231863152348178627699, 3.29497206195205231073070240784, 3.79177916484198928474186586909, 4.84800984931111549906955927248, 5.170194875391798489803193617999, 5.99985963822990787800790618762, 7.337855159413842796624373311287, 8.109723678301467489578325724994, 8.84313815728441178258059568038, 9.4071959740922641204042492077, 10.2583987028496903648207334596, 10.48731352754032952697742589532, 11.28857616202586361230337923698, 12.14164375628444314594651247310, 12.63531728977350696723304043136, 13.52049777929403112261104131593, 14.23873038674457437015910715774, 14.76995439764605162319272607161, 15.323393281644302065933871726544, 16.31710993107918815452141923607, 17.006968556519818016601150510386, 17.50264956582997629928237143489, 18.476680470321268357737275515926

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