Properties

Label 1-4235-4235.1227-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.881 + 0.472i$
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  + (−0.647 + 0.761i)2-s + (−0.207 + 0.978i)3-s + (−0.161 − 0.986i)4-s + (−0.610 − 0.791i)6-s + (0.856 + 0.516i)8-s + (−0.913 − 0.406i)9-s + (0.998 + 0.0475i)12-s + (−0.0570 − 0.998i)13-s + (−0.948 + 0.318i)16-s + (−0.846 + 0.532i)17-s + (0.901 − 0.432i)18-s + (0.640 + 0.768i)19-s + (−0.814 − 0.580i)23-s + (−0.683 + 0.730i)24-s + (0.797 + 0.603i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.647 + 0.761i)2-s + (−0.207 + 0.978i)3-s + (−0.161 − 0.986i)4-s + (−0.610 − 0.791i)6-s + (0.856 + 0.516i)8-s + (−0.913 − 0.406i)9-s + (0.998 + 0.0475i)12-s + (−0.0570 − 0.998i)13-s + (−0.948 + 0.318i)16-s + (−0.846 + 0.532i)17-s + (0.901 − 0.432i)18-s + (0.640 + 0.768i)19-s + (−0.814 − 0.580i)23-s + (−0.683 + 0.730i)24-s + (0.797 + 0.603i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.881 + 0.472i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.881 + 0.472i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1613358308 + 0.6429684452i\)
\(L(\frac12)\) \(\approx\) \(0.1613358308 + 0.6429684452i\)
\(L(1)\) \(\approx\) \(0.5218396533 + 0.3516158770i\)
\(L(1)\) \(\approx\) \(0.5218396533 + 0.3516158770i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.647 - 0.761i)T \)
3 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (0.0570 + 0.998i)T \)
17 \( 1 + (0.846 - 0.532i)T \)
19 \( 1 + (-0.640 - 0.768i)T \)
23 \( 1 + (0.814 + 0.580i)T \)
29 \( 1 + (-0.897 - 0.441i)T \)
31 \( 1 + (0.625 + 0.780i)T \)
37 \( 1 + (0.703 + 0.710i)T \)
41 \( 1 + (-0.0285 - 0.999i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + (0.901 + 0.432i)T \)
53 \( 1 + (0.318 - 0.948i)T \)
59 \( 1 + (-0.851 - 0.524i)T \)
61 \( 1 + (-0.179 - 0.983i)T \)
67 \( 1 + (-0.690 - 0.723i)T \)
71 \( 1 + (-0.696 + 0.717i)T \)
73 \( 1 + (0.917 + 0.398i)T \)
79 \( 1 + (0.905 - 0.424i)T \)
83 \( 1 + (0.491 + 0.870i)T \)
89 \( 1 + (-0.786 + 0.618i)T \)
97 \( 1 + (0.226 - 0.974i)T \)
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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

−18.05467247986952438907071761939, −17.62404497898788157821250924046, −17.12657523015132589266947310038, −16.061741971142784805573516252146, −15.82658873417704695039557173970, −14.26439431104740022167759274125, −13.8964386385614998780899524173, −13.174939162190138569699664041322, −12.53403791270231237585851872529, −11.6890842597378031859098691811, −11.493104640833005483886804573867, −10.71296867888208281146547399377, −9.729604653248776752548975762460, −9.13746244197987202490023316706, −8.4170101139357158237231642629, −7.75814625163064304660518269229, −6.84910063331531396309449710119, −6.656261824637658045373600228064, −5.33045940732826902636642469772, −4.58942043043322369298932320322, −3.59533695495224127947728757825, −2.719110728225121595766976308981, −2.040980653276866818967284970912, −1.360554671112673902092469305604, −0.334669324599853391177966991203, 0.73498459458735292798493393386, 1.93253870506809684923950161233, 2.95403933288479095060397416144, 3.97482348159460292217148318894, 4.60288137047613765574839173521, 5.559565478117394086150572776817, 5.87125791459115398441676901345, 6.75808675856547844096223889828, 7.68175468246592701735577975384, 8.39096815840430536353218775916, 8.90109994226564140848556248958, 9.78946170660153401360429822730, 10.28259978162618177856275850710, 10.79538480434909013629196329272, 11.59452370870160097988392095416, 12.49157753178890936480939891183, 13.42924865476632763188523079184, 14.290431798366204432200404755387, 14.76516095930045639480590378863, 15.43942386243758910014586467375, 16.053968814109897550208075738568, 16.46000946920346820956774686915, 17.33636349895310794751651590112, 17.78826464221971599266690643422, 18.36924042943089795015200174915

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