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Properties

Label 1-4012-4012.563-r0-0-0
Degree $1$
Conductor $4012$
Sign $-0.999 + 0.0135i$
Analytic cond. $18.6316$
Root an. cond. $18.6316$
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  + (−0.395 − 0.918i)3-s + (0.779 − 0.626i)5-s + (−0.444 − 0.895i)7-s + (−0.687 + 0.725i)9-s + (0.188 − 0.982i)11-s + (−0.725 + 0.687i)13-s + (−0.883 − 0.468i)15-s + (0.986 + 0.161i)19-s + (−0.647 + 0.762i)21-s + (0.870 − 0.492i)23-s + (0.214 − 0.976i)25-s + (0.938 + 0.344i)27-s + (0.241 − 0.970i)29-s + (−0.812 − 0.583i)31-s + (−0.976 + 0.214i)33-s + ⋯
L(s)  = 1  + (−0.395 − 0.918i)3-s + (0.779 − 0.626i)5-s + (−0.444 − 0.895i)7-s + (−0.687 + 0.725i)9-s + (0.188 − 0.982i)11-s + (−0.725 + 0.687i)13-s + (−0.883 − 0.468i)15-s + (0.986 + 0.161i)19-s + (−0.647 + 0.762i)21-s + (0.870 − 0.492i)23-s + (0.214 − 0.976i)25-s + (0.938 + 0.344i)27-s + (0.241 − 0.970i)29-s + (−0.812 − 0.583i)31-s + (−0.976 + 0.214i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $-0.999 + 0.0135i$
Analytic conductor: \(18.6316\)
Root analytic conductor: \(18.6316\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4012} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4012,\ (0:\ ),\ -0.999 + 0.0135i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009695691204 - 1.430511628i\)
\(L(\frac12)\) \(\approx\) \(0.009695691204 - 1.430511628i\)
\(L(1)\) \(\approx\) \(0.7889415951 - 0.6232199485i\)
\(L(1)\) \(\approx\) \(0.7889415951 - 0.6232199485i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
59 \( 1 \)
good3 \( 1 + (-0.395 - 0.918i)T \)
5 \( 1 + (0.779 - 0.626i)T \)
7 \( 1 + (-0.444 - 0.895i)T \)
11 \( 1 + (0.188 - 0.982i)T \)
13 \( 1 + (-0.725 + 0.687i)T \)
19 \( 1 + (0.986 + 0.161i)T \)
23 \( 1 + (0.870 - 0.492i)T \)
29 \( 1 + (0.241 - 0.970i)T \)
31 \( 1 + (-0.812 - 0.583i)T \)
37 \( 1 + (-0.744 - 0.667i)T \)
41 \( 1 + (0.492 - 0.870i)T \)
43 \( 1 + (0.827 + 0.561i)T \)
47 \( 1 + (0.994 - 0.108i)T \)
53 \( 1 + (0.605 + 0.796i)T \)
61 \( 1 + (-0.241 - 0.970i)T \)
67 \( 1 + (0.0541 - 0.998i)T \)
71 \( 1 + (0.626 - 0.779i)T \)
73 \( 1 + (0.996 + 0.0811i)T \)
79 \( 1 + (-0.395 + 0.918i)T \)
83 \( 1 + (-0.419 + 0.907i)T \)
89 \( 1 + (-0.856 + 0.515i)T \)
97 \( 1 + (-0.0811 - 0.996i)T \)
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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

−18.62232769228928106584753502757, −17.95343785244424485204669690392, −17.52275031970755931642650368506, −16.87506433774845351442274445971, −15.93410658669219368618272630671, −15.455517150436734438446467460854, −14.72536059846915978920452866014, −14.41873855217378060169362278795, −13.29841700404742762099783456669, −12.54463636967517057090669662820, −11.955025929064044916672191825118, −11.16485508597241078522702279079, −10.34255397938234571976039405152, −9.90700573527357168369114294543, −9.25642610454977937135035913665, −8.776208546002651559140988747948, −7.376758970728600286249782450822, −6.88184794438414633068930267274, −5.94775252783072460068228919357, −5.27831729533025665625323207397, −4.95062501624667548234579982567, −3.666434034101592848394800823379, −2.97262718827952646908298003027, −2.40294523907445702904037257333, −1.22624254206665315017036182830, 0.48758086562347594677581019861, 1.04778222097924926610561362393, 2.02565688689544615897563582815, 2.78248976223339072049453641521, 3.84845020810121146223740558399, 4.752762509073268589659960406936, 5.5912337713613357771258127034, 6.0953026283954300101067164214, 6.94604372658031633238772167772, 7.45052619974797330134965487182, 8.33482278670891219445680661475, 9.19192883357436688944152057377, 9.69334593619319360066888487650, 10.746914152509404302245754881296, 11.18178049591971216423872835612, 12.24126923203398892053243270415, 12.58525346917509751046536394299, 13.450197326337487595737761464341, 14.00000175196181004788180357139, 14.18711402287092571650280232879, 15.61769698103089738270232827592, 16.52387862669397386654545654322, 16.84395974884816310153122783736, 17.265138187370787843998973910553, 18.12957703927664885232229647604

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