Properties

Label 1-4235-4235.2939-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: $\chi_{4235} (2939, \cdot )$
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

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

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