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Properties

Label 1-4235-4235.1024-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.320 + 0.947i$
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  + (−0.0475 + 0.998i)2-s + (0.5 − 0.866i)3-s + (−0.995 − 0.0950i)4-s + (0.841 + 0.540i)6-s + (0.142 − 0.989i)8-s + (−0.5 − 0.866i)9-s + (−0.580 + 0.814i)12-s + (−0.415 − 0.909i)13-s + (0.981 + 0.189i)16-s + (−0.235 + 0.971i)17-s + (0.888 − 0.458i)18-s + (0.235 + 0.971i)19-s + (−0.981 − 0.189i)23-s + (−0.786 − 0.618i)24-s + (0.928 − 0.371i)26-s − 27-s + ⋯
L(s)  = 1  + (−0.0475 + 0.998i)2-s + (0.5 − 0.866i)3-s + (−0.995 − 0.0950i)4-s + (0.841 + 0.540i)6-s + (0.142 − 0.989i)8-s + (−0.5 − 0.866i)9-s + (−0.580 + 0.814i)12-s + (−0.415 − 0.909i)13-s + (0.981 + 0.189i)16-s + (−0.235 + 0.971i)17-s + (0.888 − 0.458i)18-s + (0.235 + 0.971i)19-s + (−0.981 − 0.189i)23-s + (−0.786 − 0.618i)24-s + (0.928 − 0.371i)26-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049317995 + 0.7528607326i\)
\(L(\frac12)\) \(\approx\) \(1.049317995 + 0.7528607326i\)
\(L(1)\) \(\approx\) \(0.9621333879 + 0.2179392041i\)
\(L(1)\) \(\approx\) \(0.9621333879 + 0.2179392041i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0475 + 0.998i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (-0.235 + 0.971i)T \)
19 \( 1 + (0.235 + 0.971i)T \)
23 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (0.580 + 0.814i)T \)
37 \( 1 + (0.995 - 0.0950i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.888 + 0.458i)T \)
53 \( 1 + (-0.981 + 0.189i)T \)
59 \( 1 + (0.0475 + 0.998i)T \)
61 \( 1 + (-0.888 - 0.458i)T \)
67 \( 1 + (0.888 - 0.458i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.327 + 0.945i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−18.40963617255478422816476023292, −17.63123411039011093993985055165, −16.92481153850593474849168277461, −16.21319309792986917205579426105, −15.5365951088122390180350207932, −14.61999282701788695571259934426, −14.14584450785299674346308541615, −13.50248015954735261019073127695, −12.87296464007854363129880622717, −11.69962382852027155122082519489, −11.4969085976073562299353940365, −10.69892968486229923407423415276, −9.89422822621386281335283999616, −9.26647571475475006721623472458, −9.07546801048200707966053195623, −7.95799953949214874096277836684, −7.40358393431949129112288928401, −6.125777170508150130706849903667, −5.19449552495877556636238096596, −4.50914641063033476693507311956, −4.06749561417826948044441314143, −3.09689526078450139330729025473, −2.47845597618183157191034191146, −1.80712434453226426212333811308, −0.43158937774956032156804012368, 0.808903787661364016557134073555, 1.714696732762413221556143160590, 2.731902911174379564226416770839, 3.66381150581112411294261866459, 4.30711918324089809276390599768, 5.508798149814613431123695485075, 5.98294102528298389347878447113, 6.62149552206716343913414109546, 7.65261013885005786484245879609, 7.84091075724958786447804518651, 8.547252002823516745465993677138, 9.35055334815007779204953071425, 10.03689631768960618185158068202, 10.84545620353842867147363343598, 12.09791873337321629187727831089, 12.56055178474676341767675384155, 13.14298135474065577936707193116, 13.9322604633996895752839238467, 14.44081083293077895603679180409, 15.07617818562878701929872598662, 15.65958852068078795785256379325, 16.55166288499237610136794889022, 17.28198443483810174365073111958, 17.73510775647635665946420447492, 18.456041332687536007262608540668

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