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Properties

Label 1-4235-4235.1019-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.654 + 0.756i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (0.879 + 0.475i)2-s + (0.104 − 0.994i)3-s + (0.548 + 0.836i)4-s + (0.564 − 0.825i)6-s + (0.0855 + 0.996i)8-s + (−0.978 − 0.207i)9-s + (0.888 − 0.458i)12-s + (−0.254 + 0.967i)13-s + (−0.398 + 0.917i)16-s + (0.345 + 0.938i)17-s + (−0.761 − 0.647i)18-s + (−0.272 − 0.962i)19-s + (0.995 + 0.0950i)23-s + (0.999 + 0.0190i)24-s + (−0.683 + 0.730i)26-s + (−0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.879 + 0.475i)2-s + (0.104 − 0.994i)3-s + (0.548 + 0.836i)4-s + (0.564 − 0.825i)6-s + (0.0855 + 0.996i)8-s + (−0.978 − 0.207i)9-s + (0.888 − 0.458i)12-s + (−0.254 + 0.967i)13-s + (−0.398 + 0.917i)16-s + (0.345 + 0.938i)17-s + (−0.761 − 0.647i)18-s + (−0.272 − 0.962i)19-s + (0.995 + 0.0950i)23-s + (0.999 + 0.0190i)24-s + (−0.683 + 0.730i)26-s + (−0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.654 + 0.756i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.654 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.083448604 + 2.369089657i\)
\(L(\frac12)\) \(\approx\) \(1.083448604 + 2.369089657i\)
\(L(1)\) \(\approx\) \(1.620740269 + 0.3263201946i\)
\(L(1)\) \(\approx\) \(1.620740269 + 0.3263201946i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.879 + 0.475i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (-0.254 + 0.967i)T \)
17 \( 1 + (0.345 + 0.938i)T \)
19 \( 1 + (-0.272 - 0.962i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (0.466 + 0.884i)T \)
31 \( 1 + (0.161 - 0.986i)T \)
37 \( 1 + (0.625 + 0.780i)T \)
41 \( 1 + (-0.610 - 0.791i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (0.761 - 0.647i)T \)
53 \( 1 + (0.398 + 0.917i)T \)
59 \( 1 + (-0.991 - 0.132i)T \)
61 \( 1 + (0.851 + 0.524i)T \)
67 \( 1 + (-0.235 - 0.971i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (0.00951 + 0.999i)T \)
79 \( 1 + (-0.797 + 0.603i)T \)
83 \( 1 + (-0.736 - 0.676i)T \)
89 \( 1 + (0.928 + 0.371i)T \)
97 \( 1 + (-0.516 + 0.856i)T \)
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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

−17.99666623101673446343673150310, −17.10157020066973785995387672174, −16.342045164851870882386171883200, −15.8269837402679876107482499406, −15.08588462612116389391253269195, −14.61248073690876709808881413948, −13.99980955766251381752831275933, −13.21927026437353157819489999371, −12.49369774885231024268944371291, −11.76647314435688623547676472904, −11.137649055152311423053299898, −10.39450754325998157671535189553, −9.94265754782006082331509145549, −9.26593872786691887998516137945, −8.29141282000717114450916062989, −7.50153839608262856437999180486, −6.46937435692435035439627149868, −5.72767334049492341414936867490, −5.05764885415864094518013420161, −4.553382250413775803609098948385, −3.62904745136348780754274843748, −3.0241193862191288122472938433, −2.44373329321655202489538164798, −1.20912440307834948812822283886, −0.26016093523926509263456001294, 1.056425135460303636623358377287, 1.99173742503176531022146332551, 2.6477416519474098224949980616, 3.478630932041627790685396706264, 4.33622767718051072186702497408, 5.16567272722709261314541003692, 5.879903205262508684220714909725, 6.75803630075670115021027899999, 6.975263572902220295008541055451, 7.847155340871289956732162342634, 8.60734685269640987799112281159, 9.12283423920087443608445349114, 10.43346348291985421616914810488, 11.28236861047410920733062876370, 11.8017591206984504447780417800, 12.50836702511384095959859644550, 13.06917987617958692003113901184, 13.70692462044416009008666696779, 14.23298621516753585495734818153, 15.05713195133106690283714344964, 15.415181825511780945750873957779, 16.66329738933132639061495393625, 16.980294398288381719620109029207, 17.527986383561893262414769571335, 18.5762830851598078622791642089

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