Properties

Label 1-4235-4235.1069-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.976 - 0.215i$
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  + (−0.625 + 0.780i)2-s + (0.669 + 0.743i)3-s + (−0.217 − 0.976i)4-s + (−0.998 + 0.0570i)6-s + (0.897 + 0.441i)8-s + (−0.104 + 0.994i)9-s + (0.580 − 0.814i)12-s + (−0.198 + 0.980i)13-s + (−0.905 + 0.424i)16-s + (−0.380 − 0.924i)17-s + (−0.710 − 0.703i)18-s + (0.997 + 0.0760i)19-s + (−0.981 − 0.189i)23-s + (0.272 + 0.962i)24-s + (−0.640 − 0.768i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.625 + 0.780i)2-s + (0.669 + 0.743i)3-s + (−0.217 − 0.976i)4-s + (−0.998 + 0.0570i)6-s + (0.897 + 0.441i)8-s + (−0.104 + 0.994i)9-s + (0.580 − 0.814i)12-s + (−0.198 + 0.980i)13-s + (−0.905 + 0.424i)16-s + (−0.380 − 0.924i)17-s + (−0.710 − 0.703i)18-s + (0.997 + 0.0760i)19-s + (−0.981 − 0.189i)23-s + (0.272 + 0.962i)24-s + (−0.640 − 0.768i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.976 - 0.215i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.976 - 0.215i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1250941705 + 1.149298658i\)
\(L(\frac12)\) \(\approx\) \(-0.1250941705 + 1.149298658i\)
\(L(1)\) \(\approx\) \(0.6685387009 + 0.6054256711i\)
\(L(1)\) \(\approx\) \(0.6685387009 + 0.6054256711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.625 + 0.780i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (-0.198 + 0.980i)T \)
17 \( 1 + (-0.380 - 0.924i)T \)
19 \( 1 + (0.997 + 0.0760i)T \)
23 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 + (0.0285 + 0.999i)T \)
31 \( 1 + (-0.00951 + 0.999i)T \)
37 \( 1 + (0.398 + 0.917i)T \)
41 \( 1 + (0.774 - 0.633i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.710 + 0.703i)T \)
53 \( 1 + (0.905 + 0.424i)T \)
59 \( 1 + (0.935 - 0.353i)T \)
61 \( 1 + (0.988 - 0.151i)T \)
67 \( 1 + (0.888 - 0.458i)T \)
71 \( 1 + (0.941 - 0.336i)T \)
73 \( 1 + (-0.797 + 0.603i)T \)
79 \( 1 + (-0.999 + 0.0380i)T \)
83 \( 1 + (-0.974 - 0.226i)T \)
89 \( 1 + (-0.723 - 0.690i)T \)
97 \( 1 + (0.696 - 0.717i)T \)
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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

−18.054116609501375199591501607229, −17.65959615113207906738315511272, −16.98146387208146248848254766694, −16.00586124534186341492386912009, −15.309527553818115097588081670266, −14.50553769627643169448391778708, −13.70058002235167594246399075917, −13.04602331869572696662176094749, −12.69229737340622297665764860558, −11.732351935854605605522256080748, −11.39090846075925875418393063238, −10.17117782652826060111645276709, −9.8788120054643032761763850775, −8.97916588575505345733681424501, −8.27956866404839451059219679109, −7.790094544678021585124106695577, −7.193122677648234426297227919001, −6.21555101671337767275231466092, −5.38575584415425830040892288434, −4.00217624485721938575796170395, −3.68989485029069260191250307189, −2.56916995184680630790578863362, −2.22341586880941204095878123389, −1.20707698585466218381067444933, −0.38848684870886791238127524107, 1.179450552189969416871764870348, 2.09871757484837636400767135026, 2.95444370543965907065084772912, 3.9844275174232768575649824685, 4.7584972172854318599031303380, 5.25323786637136394970216855339, 6.25394889270644719841407146676, 7.09972482576936887369898010754, 7.61388585741660596173175702301, 8.56055221452130142742224531829, 8.934923450412744437828123041324, 9.88008151025624591920107131613, 9.95133878200234596890507271400, 11.142106187095957134350220702568, 11.56578879892372881225763013766, 12.79560859921378252373101848807, 13.7612115022097515468302455738, 14.25395067414261964643111310478, 14.54994456403618784379560832702, 15.68604807919248364362664748169, 15.96692080977500656935632591248, 16.48199351643806909898481968477, 17.24685879354535731436909664582, 18.18316914828732841565404980873, 18.51732170714457129387080277288

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