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Properties

Label 1-4235-4235.1048-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.982 - 0.185i$
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  + (0.703 + 0.710i)2-s + (−0.994 + 0.104i)3-s + (−0.00951 + 0.999i)4-s + (−0.774 − 0.633i)6-s + (−0.717 + 0.696i)8-s + (0.978 − 0.207i)9-s + (−0.0950 − 0.995i)12-s + (−0.676 + 0.736i)13-s + (−0.999 − 0.0190i)16-s + (−0.475 + 0.879i)17-s + (0.836 + 0.548i)18-s + (−0.991 − 0.132i)19-s + (−0.945 − 0.327i)23-s + (0.640 − 0.768i)24-s + (−0.999 + 0.0380i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.703 + 0.710i)2-s + (−0.994 + 0.104i)3-s + (−0.00951 + 0.999i)4-s + (−0.774 − 0.633i)6-s + (−0.717 + 0.696i)8-s + (0.978 − 0.207i)9-s + (−0.0950 − 0.995i)12-s + (−0.676 + 0.736i)13-s + (−0.999 − 0.0190i)16-s + (−0.475 + 0.879i)17-s + (0.836 + 0.548i)18-s + (−0.991 − 0.132i)19-s + (−0.945 − 0.327i)23-s + (0.640 − 0.768i)24-s + (−0.999 + 0.0380i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.982 - 0.185i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.982 - 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5817321851 - 0.05430202026i\)
\(L(\frac12)\) \(\approx\) \(0.5817321851 - 0.05430202026i\)
\(L(1)\) \(\approx\) \(0.7402699199 + 0.4358107718i\)
\(L(1)\) \(\approx\) \(0.7402699199 + 0.4358107718i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.703 + 0.710i)T \)
3 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (-0.676 + 0.736i)T \)
17 \( 1 + (-0.475 + 0.879i)T \)
19 \( 1 + (-0.991 - 0.132i)T \)
23 \( 1 + (-0.945 - 0.327i)T \)
29 \( 1 + (-0.941 - 0.336i)T \)
31 \( 1 + (0.398 - 0.917i)T \)
37 \( 1 + (0.318 + 0.948i)T \)
41 \( 1 + (0.362 - 0.931i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + (-0.836 + 0.548i)T \)
53 \( 1 + (-0.0190 - 0.999i)T \)
59 \( 1 + (0.988 + 0.151i)T \)
61 \( 1 + (-0.964 + 0.263i)T \)
67 \( 1 + (0.998 - 0.0475i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (-0.424 + 0.905i)T \)
79 \( 1 + (0.0665 + 0.997i)T \)
83 \( 1 + (-0.389 + 0.921i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (0.170 - 0.985i)T \)
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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

−18.35067643869913933685272174050, −17.828072959403659136113162027035, −17.192100034646894573654899458964, −16.15867144972961296040589596177, −15.784571817636366765207502551407, −14.87373850684499277529649830698, −14.31952573385483720724362447039, −13.31951666619756097065681002733, −12.88369232550568999292769446777, −12.22233660068842730008054752146, −11.647914203023987908028429816325, −10.94359660156337189063823822963, −10.39994724150824963945534915080, −9.74231771573149626928946509008, −8.99428929547882323014011843342, −7.76049809539539014567626259159, −7.05457266349413025200200315000, −6.208815207017100115162648421885, −5.67739247221272658308465068249, −4.88448102730578051793515554613, −4.38348803723572537321751275566, −3.483804955919235883836358146310, −2.49537049106828212333545988967, −1.79375084553480837817777702874, −0.75795376380037231434099316296, 0.18330620049862122192814492583, 1.80844050153093558317454611751, 2.50363698247211437118965031723, 3.97949038060206309117473666978, 4.154312522676050829959393528140, 4.97281449021963118667530820901, 5.82935445541500184145994136100, 6.32932528094733208989900577841, 6.92851062948253340946045629725, 7.73200551172427487010111411254, 8.47864622554667159045729376728, 9.406835871893728456580784117722, 10.14695862146834969679513058118, 11.1172659879825473870041999836, 11.55779316120920359443155884507, 12.40604681726116495344063221297, 12.812476165791423101572700462393, 13.552088188908206350720327891823, 14.4196850376982460555818285839, 15.07855650864450775317859582469, 15.60372093426713224485069192180, 16.41839410651758098673268342542, 16.947931087952484350462176198882, 17.36514771660608467497674493302, 18.07331929436487017194242512010

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