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Properties

Label 1-4235-4235.249-r1-0-0
Degree $1$
Conductor $4235$
Sign $0.120 - 0.992i$
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  + (0.797 − 0.603i)2-s + (0.104 − 0.994i)3-s + (0.272 − 0.962i)4-s + (−0.516 − 0.856i)6-s + (−0.362 − 0.931i)8-s + (−0.978 − 0.207i)9-s + (−0.928 − 0.371i)12-s + (0.897 − 0.441i)13-s + (−0.851 − 0.524i)16-s + (−0.217 + 0.976i)17-s + (−0.905 + 0.424i)18-s + (−0.749 − 0.662i)19-s + (−0.235 + 0.971i)23-s + (−0.964 + 0.263i)24-s + (0.449 − 0.893i)26-s + (−0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.797 − 0.603i)2-s + (0.104 − 0.994i)3-s + (0.272 − 0.962i)4-s + (−0.516 − 0.856i)6-s + (−0.362 − 0.931i)8-s + (−0.978 − 0.207i)9-s + (−0.928 − 0.371i)12-s + (0.897 − 0.441i)13-s + (−0.851 − 0.524i)16-s + (−0.217 + 0.976i)17-s + (−0.905 + 0.424i)18-s + (−0.749 − 0.662i)19-s + (−0.235 + 0.971i)23-s + (−0.964 + 0.263i)24-s + (0.449 − 0.893i)26-s + (−0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.120 - 0.992i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ 0.120 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.530166309 - 2.241225362i\)
\(L(\frac12)\) \(\approx\) \(2.530166309 - 2.241225362i\)
\(L(1)\) \(\approx\) \(1.243189662 - 1.025821299i\)
\(L(1)\) \(\approx\) \(1.243189662 - 1.025821299i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.797 - 0.603i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (0.897 - 0.441i)T \)
17 \( 1 + (-0.217 + 0.976i)T \)
19 \( 1 + (-0.749 - 0.662i)T \)
23 \( 1 + (-0.235 + 0.971i)T \)
29 \( 1 + (0.870 + 0.491i)T \)
31 \( 1 + (0.640 + 0.768i)T \)
37 \( 1 + (-0.999 - 0.0380i)T \)
41 \( 1 + (-0.974 + 0.226i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (0.905 + 0.424i)T \)
53 \( 1 + (0.851 - 0.524i)T \)
59 \( 1 + (-0.290 - 0.956i)T \)
61 \( 1 + (-0.123 + 0.992i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.991 - 0.132i)T \)
79 \( 1 + (0.935 - 0.353i)T \)
83 \( 1 + (-0.564 + 0.825i)T \)
89 \( 1 + (0.580 + 0.814i)T \)
97 \( 1 + (0.254 - 0.967i)T \)
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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

−18.27799152758439369315571982638, −17.27365077296542733569181816428, −16.85923433768524708467646489124, −16.11741286038403129923090263640, −15.648694887214686999184234573322, −15.10374050766578916070063481170, −14.218323297448659321186932783894, −13.87511095241060193896749582149, −13.18687467881250764409229917833, −12.01827088867440301088124715246, −11.80674143157250673968713071529, −10.7471140081966227914732045970, −10.30445987055639142968794418229, −9.16283401450482325527384049759, −8.64389593046173013315115430267, −8.0430695227165122624091463566, −7.035771351361105403257943886542, −6.25249611601452565777497956401, −5.71533621137390062494039099279, −4.78990388760914780907451360304, −4.257315942765318146725571089758, −3.66098761450553487569406612717, −2.77171209948060791939790544740, −2.07659578834343369121639133424, −0.450682942975302056983474288060, 0.701499390622446180610927787499, 1.42693990837489723045396253476, 2.10079093019292892784747070270, 3.02044564609317409261402224105, 3.5854960800535494472677246617, 4.54213308652197713472778176985, 5.43269607226434930905741852105, 6.13486426314364924348619110074, 6.6334445269762189220323544088, 7.43739017381965059904072329069, 8.48556851017969775098028359704, 8.86540668397185677134747385992, 10.04642901679885600597248844595, 10.76258590567299105427572063598, 11.26816374655352318989650376544, 12.16304101672779983907219351467, 12.56174777644215970690450662294, 13.356916746124860460006506256879, 13.6971128697130047151157957886, 14.426983329459126551070968702827, 15.27302060317570226000368288555, 15.68548503161185348889090704636, 16.75726080110625749163328239338, 17.77131878011418180477790901610, 17.94605572094605348755633992808

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