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Properties

Label 1-4235-4235.3447-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: $\chi_{4235} (3447, \cdot )$
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

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

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