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Properties

Label 1-4235-4235.1917-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.975 - 0.220i$
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  + (−0.999 − 0.0285i)2-s + (−0.587 − 0.809i)3-s + (0.998 + 0.0570i)4-s + (0.564 + 0.825i)6-s + (−0.996 − 0.0855i)8-s + (−0.309 + 0.951i)9-s + (−0.540 − 0.841i)12-s + (0.967 − 0.254i)13-s + (0.993 + 0.113i)16-s + (0.170 + 0.985i)17-s + (0.336 − 0.941i)18-s + (0.696 + 0.717i)19-s + (−0.909 − 0.415i)23-s + (0.516 + 0.856i)24-s + (−0.974 + 0.226i)26-s + (0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.999 − 0.0285i)2-s + (−0.587 − 0.809i)3-s + (0.998 + 0.0570i)4-s + (0.564 + 0.825i)6-s + (−0.996 − 0.0855i)8-s + (−0.309 + 0.951i)9-s + (−0.540 − 0.841i)12-s + (0.967 − 0.254i)13-s + (0.993 + 0.113i)16-s + (0.170 + 0.985i)17-s + (0.336 − 0.941i)18-s + (0.696 + 0.717i)19-s + (−0.909 − 0.415i)23-s + (0.516 + 0.856i)24-s + (−0.974 + 0.226i)26-s + (0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.975 - 0.220i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1917, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.975 - 0.220i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8505046238 - 0.09511278420i\)
\(L(\frac12)\) \(\approx\) \(0.8505046238 - 0.09511278420i\)
\(L(1)\) \(\approx\) \(0.6132194526 - 0.1099376497i\)
\(L(1)\) \(\approx\) \(0.6132194526 - 0.1099376497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.999 - 0.0285i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (0.967 - 0.254i)T \)
17 \( 1 + (0.170 + 0.985i)T \)
19 \( 1 + (0.696 + 0.717i)T \)
23 \( 1 + (-0.909 - 0.415i)T \)
29 \( 1 + (0.466 - 0.884i)T \)
31 \( 1 + (-0.774 + 0.633i)T \)
37 \( 1 + (0.931 - 0.362i)T \)
41 \( 1 + (-0.610 + 0.791i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + (-0.336 - 0.941i)T \)
53 \( 1 + (-0.113 - 0.993i)T \)
59 \( 1 + (0.610 + 0.791i)T \)
61 \( 1 + (0.0285 + 0.999i)T \)
67 \( 1 + (-0.281 - 0.959i)T \)
71 \( 1 + (0.897 - 0.441i)T \)
73 \( 1 + (-0.491 - 0.870i)T \)
79 \( 1 + (0.921 - 0.389i)T \)
83 \( 1 + (-0.676 - 0.736i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.856 - 0.516i)T \)
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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

−18.336293978181302933236676317193, −17.69074123303900443818521056011, −17.097583475483650200162573662618, −16.263791763035279231552369985539, −15.91433123414072637786451741510, −15.46089709624002498991890672346, −14.472362353689687260057758037587, −13.83618667780344630704898873089, −12.69970002360525087464235534846, −11.89982651659589674167166155231, −11.30737544302558320453784896480, −10.9154221243975266817770610109, −10.03371310033115167653896732169, −9.45018991257658699933120609198, −8.96699427541933815718625196359, −8.12743587290422582789263584866, −7.23947448429810391057714455776, −6.58759443258830690920516123161, −5.77423210788408989893660693438, −5.21276862888685586195649156859, −4.13557913976096194711198629273, −3.35614778131095360703151882987, −2.56691010317530783173396524063, −1.388797087527916792294296804964, −0.5680629919659043080947460497, 0.70038746879999116144433862084, 1.48410665074825687926733654107, 2.11334325904133725240217041792, 3.13440268288323667321059607698, 4.0088870950638358683654383520, 5.303260072193644393764082714703, 6.105896430531036337203302677465, 6.34778646562664763855442655861, 7.35869354099761572773093121279, 8.06062934946928644611563169554, 8.36192073749204350480470799924, 9.38374598403969914144298124731, 10.305539993437528668430152163956, 10.66110883610803994680546353021, 11.560594512969478337269748059613, 11.9884244867148991776481029086, 12.76989817973640729834810852382, 13.405071663162843998974287624360, 14.34734441666248281908932216224, 15.06215330280551788268412443550, 16.13829051035634240003584571909, 16.333803448796654607910297114114, 17.10541717003911859660825052247, 18.00270193473102970218559725065, 18.11053946653143560980509501806

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