↑

Properties

Label 1-4012-4012.451-r0-0-0
Degree $1$
Conductor $4012$
Sign $-0.330 - 0.943i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0270 + 0.999i)3-s + (−0.538 − 0.842i)5-s + (0.990 − 0.135i)7-s + (−0.998 + 0.0541i)9-s + (0.395 + 0.918i)11-s + (0.0541 − 0.998i)13-s + (0.827 − 0.561i)15-s + (0.319 − 0.947i)19-s + (0.161 + 0.986i)21-s + (−0.970 + 0.241i)23-s + (−0.419 + 0.907i)25-s + (−0.0811 − 0.996i)27-s + (−0.955 + 0.293i)29-s + (−0.895 − 0.444i)31-s + (−0.907 + 0.419i)33-s + ⋯
L(s)  = 1  + (0.0270 + 0.999i)3-s + (−0.538 − 0.842i)5-s + (0.990 − 0.135i)7-s + (−0.998 + 0.0541i)9-s + (0.395 + 0.918i)11-s + (0.0541 − 0.998i)13-s + (0.827 − 0.561i)15-s + (0.319 − 0.947i)19-s + (0.161 + 0.986i)21-s + (−0.970 + 0.241i)23-s + (−0.419 + 0.907i)25-s + (−0.0811 − 0.996i)27-s + (−0.955 + 0.293i)29-s + (−0.895 − 0.444i)31-s + (−0.907 + 0.419i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $-0.330 - 0.943i$
Analytic conductor: \(18.6316\)
Root analytic conductor: \(18.6316\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4012} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4012,\ (0:\ ),\ -0.330 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4105821347 - 0.5786075879i\)
\(L(\frac12)\) \(\approx\) \(0.4105821347 - 0.5786075879i\)
\(L(1)\) \(\approx\) \(0.9061253278 + 0.04936542788i\)
\(L(1)\) \(\approx\) \(0.9061253278 + 0.04936542788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
59 \( 1 \)
good3 \( 1 + (0.0270 + 0.999i)T \)
5 \( 1 + (-0.538 - 0.842i)T \)
7 \( 1 + (0.990 - 0.135i)T \)
11 \( 1 + (0.395 + 0.918i)T \)
13 \( 1 + (0.0541 - 0.998i)T \)
19 \( 1 + (0.319 - 0.947i)T \)
23 \( 1 + (-0.970 + 0.241i)T \)
29 \( 1 + (-0.955 + 0.293i)T \)
31 \( 1 + (-0.895 - 0.444i)T \)
37 \( 1 + (0.779 + 0.626i)T \)
41 \( 1 + (0.241 - 0.970i)T \)
43 \( 1 + (0.928 - 0.370i)T \)
47 \( 1 + (-0.976 + 0.214i)T \)
53 \( 1 + (-0.963 - 0.267i)T \)
61 \( 1 + (0.955 + 0.293i)T \)
67 \( 1 + (-0.994 - 0.108i)T \)
71 \( 1 + (-0.842 - 0.538i)T \)
73 \( 1 + (-0.812 + 0.583i)T \)
79 \( 1 + (0.0270 - 0.999i)T \)
83 \( 1 + (0.762 - 0.647i)T \)
89 \( 1 + (0.468 - 0.883i)T \)
97 \( 1 + (0.583 - 0.812i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.808941023612289405023449766260, −18.04888384750619578677368622823, −17.65775410537972825413255127966, −16.46126269369244510526891676266, −16.2888430054539958463766604425, −14.933576631462657331438097522934, −14.40297174170682920163801123819, −14.166183366429324095620190996523, −13.32575967328782397858739730776, −12.34020728368242984778010360979, −11.74151110506531857358904872404, −11.25829126555551107917767545418, −10.79400640930138390624862699842, −9.57426599110280181818750343053, −8.77077572446849970715381105107, −7.91790561707236172525988418216, −7.71708713346594715082875188828, −6.75346057427844417870884607526, −6.12192951502507332407160674533, −5.501050746477645820513824345768, −4.25444907725293602538021035282, −3.64251317745320043560515111763, −2.68862070613446024573185102382, −1.89206964764740806017541992302, −1.208375873126335329775075843408, 0.19964840553157094519762250498, 1.37623499713090728000540536020, 2.29794151874589613760897553125, 3.42723039623602844466087324274, 4.117083871151979374245555562737, 4.716252648543363604965243628286, 5.2733832206849614278335076474, 5.9886542565648096385418051789, 7.449612886909123802148865458910, 7.74190259703196792187545973535, 8.67644289362727895538825191357, 9.19588644500753738856481014234, 9.92143001738598638096500585909, 10.725784267711374231460552973791, 11.42129493858704807499871168032, 11.889730020237929678589711516474, 12.77790569860026501665191240174, 13.47269980019970897051111933769, 14.51467676873417197306228742725, 14.89836669501971114317112930261, 15.579186397998380196170168765931, 16.13098591387056812477167806995, 16.899454226827730372195087472098, 17.62329008654636430459198088821, 17.84976800937516176451653214572

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