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Properties

Label 1-4235-4235.983-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.478 - 0.877i$
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  + (−0.938 − 0.345i)2-s + (0.994 + 0.104i)3-s + (0.761 + 0.647i)4-s + (−0.897 − 0.441i)6-s + (−0.491 − 0.870i)8-s + (0.978 + 0.207i)9-s + (0.690 + 0.723i)12-s + (−0.999 + 0.0285i)13-s + (0.161 + 0.986i)16-s + (0.875 + 0.483i)17-s + (−0.846 − 0.532i)18-s + (−0.905 − 0.424i)19-s + (−0.458 − 0.888i)23-s + (−0.398 − 0.917i)24-s + (0.948 + 0.318i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.938 − 0.345i)2-s + (0.994 + 0.104i)3-s + (0.761 + 0.647i)4-s + (−0.897 − 0.441i)6-s + (−0.491 − 0.870i)8-s + (0.978 + 0.207i)9-s + (0.690 + 0.723i)12-s + (−0.999 + 0.0285i)13-s + (0.161 + 0.986i)16-s + (0.875 + 0.483i)17-s + (−0.846 − 0.532i)18-s + (−0.905 − 0.424i)19-s + (−0.458 − 0.888i)23-s + (−0.398 − 0.917i)24-s + (0.948 + 0.318i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.478 - 0.877i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (983, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.478 - 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.249399093 - 0.7417999643i\)
\(L(\frac12)\) \(\approx\) \(1.249399093 - 0.7417999643i\)
\(L(1)\) \(\approx\) \(0.9569220740 - 0.1646450033i\)
\(L(1)\) \(\approx\) \(0.9569220740 - 0.1646450033i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.938 - 0.345i)T \)
3 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.999 + 0.0285i)T \)
17 \( 1 + (0.875 + 0.483i)T \)
19 \( 1 + (-0.905 - 0.424i)T \)
23 \( 1 + (-0.458 - 0.888i)T \)
29 \( 1 + (-0.974 - 0.226i)T \)
31 \( 1 + (0.432 - 0.901i)T \)
37 \( 1 + (0.924 - 0.380i)T \)
41 \( 1 + (-0.696 + 0.717i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + (0.846 - 0.532i)T \)
53 \( 1 + (0.986 + 0.161i)T \)
59 \( 1 + (0.272 - 0.962i)T \)
61 \( 1 + (-0.640 + 0.768i)T \)
67 \( 1 + (-0.371 - 0.928i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (0.836 - 0.548i)T \)
79 \( 1 + (0.217 + 0.976i)T \)
83 \( 1 + (-0.967 + 0.254i)T \)
89 \( 1 + (-0.327 - 0.945i)T \)
97 \( 1 + (-0.113 + 0.993i)T \)
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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

−18.57887961773836336045976200961, −17.91781280679823730613724500015, −17.091624998618506245403028858323, −16.574110265146628677693808670643, −15.7000100885186517265004866882, −15.19785170384537583788113619695, −14.46044159553242052531072733244, −14.09728746716457320872605738209, −13.1064797327053764808414335630, −12.246213467948561584490168925385, −11.716130642504475545058674020821, −10.53008841352178778289609887367, −10.12198004967286018705253136454, −9.37802470072397282433836583171, −8.87042857882649012744415576098, −8.04090020497954499289380784784, −7.47434695580149182850574805533, −7.02833301727117550646267504679, −6.00879774289116389679974409985, −5.25167097097795030247022495667, −4.240918912104291991860762661926, −3.27453330861100826572630524667, −2.48970117503194546074207926486, −1.82360999131034064935245648163, −0.93028203551162338751747250017, 0.53122139273964454722089452721, 1.685175727564563292325137757109, 2.3763992582211166294636040305, 2.90276805423156800729114971109, 3.936919803894395596785611222903, 4.4456240602407503730392092059, 5.769252509604021765573068639080, 6.65373625660774120080913544855, 7.454213733374142019130034337880, 7.94093769113998504078120887580, 8.596807255731767386249181341416, 9.33597617682540616563694929956, 9.90127209946473299837425914627, 10.44792782623175493823675598606, 11.25310229338311705319024955936, 12.194350859090791909950783442922, 12.697258514600265371259063527923, 13.38113255440977117896478330494, 14.38262815584976231083968896939, 15.01212919251546860212743633127, 15.406287463312656565781647347461, 16.60601227537080348655105251141, 16.723010251792848447660890685252, 17.66925968267334955247079648629, 18.529201895296009708293270027896

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