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Properties

Label 1-4029-4029.2303-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.938 - 0.345i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (−0.239 − 0.970i)2-s + (−0.885 + 0.464i)4-s + (0.180 − 0.983i)5-s + (0.998 − 0.0603i)7-s + (0.663 + 0.748i)8-s + (−0.998 + 0.0603i)10-s + (−0.180 − 0.983i)11-s + (−0.885 + 0.464i)13-s + (−0.297 − 0.954i)14-s + (0.568 − 0.822i)16-s + (−0.935 + 0.354i)19-s + (0.297 + 0.954i)20-s + (−0.911 + 0.410i)22-s + (0.707 − 0.707i)23-s + (−0.935 − 0.354i)25-s + (0.663 + 0.748i)26-s + ⋯
L(s)  = 1  + (−0.239 − 0.970i)2-s + (−0.885 + 0.464i)4-s + (0.180 − 0.983i)5-s + (0.998 − 0.0603i)7-s + (0.663 + 0.748i)8-s + (−0.998 + 0.0603i)10-s + (−0.180 − 0.983i)11-s + (−0.885 + 0.464i)13-s + (−0.297 − 0.954i)14-s + (0.568 − 0.822i)16-s + (−0.935 + 0.354i)19-s + (0.297 + 0.954i)20-s + (−0.911 + 0.410i)22-s + (0.707 − 0.707i)23-s + (−0.935 − 0.354i)25-s + (0.663 + 0.748i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.938 - 0.345i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.938 - 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2597908168 - 1.457083046i\)
\(L(\frac12)\) \(\approx\) \(0.2597908168 - 1.457083046i\)
\(L(1)\) \(\approx\) \(0.7344828069 - 0.6581431994i\)
\(L(1)\) \(\approx\) \(0.7344828069 - 0.6581431994i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.239 - 0.970i)T \)
5 \( 1 + (0.180 - 0.983i)T \)
7 \( 1 + (0.998 - 0.0603i)T \)
11 \( 1 + (-0.180 - 0.983i)T \)
13 \( 1 + (-0.885 + 0.464i)T \)
19 \( 1 + (-0.935 + 0.354i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (0.787 - 0.616i)T \)
31 \( 1 + (0.855 + 0.517i)T \)
37 \( 1 + (0.410 - 0.911i)T \)
41 \( 1 + (0.983 + 0.180i)T \)
43 \( 1 + (0.822 - 0.568i)T \)
47 \( 1 + (0.354 - 0.935i)T \)
53 \( 1 + (0.663 + 0.748i)T \)
59 \( 1 + (-0.464 + 0.885i)T \)
61 \( 1 + (0.911 + 0.410i)T \)
67 \( 1 + (-0.970 - 0.239i)T \)
71 \( 1 + (0.998 + 0.0603i)T \)
73 \( 1 + (0.297 + 0.954i)T \)
83 \( 1 + (0.464 + 0.885i)T \)
89 \( 1 + (-0.748 - 0.663i)T \)
97 \( 1 + (0.410 - 0.911i)T \)
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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

−18.59410118588823230445516513084, −17.72002817600664796636161758165, −17.57125178706634691650861665622, −17.03647749935212808974427419505, −15.828094384080551095944370327245, −15.25914838347937805787336771847, −14.7329603488962413556680356019, −14.400321265777515064800302709484, −13.5056353236553728648749948116, −12.79267009415399244275687490035, −11.92133912188833369247670413496, −10.95244558956762947727910145208, −10.4401012415620894350696270202, −9.70443737756230065256223467495, −9.063068214430386464704620616507, −7.91358636882582503838452365297, −7.72860610426442315285049847715, −6.86794999452999243166588545508, −6.31922768660937536094303373184, −5.30986882391879850784580982895, −4.776623418601012104451665458474, −4.06964520560140509035265029661, −2.8080026997994097859914877445, −2.103442771931126669195676415252, −1.01940647055461130628801579530, 0.5554475294323462097418662733, 1.20723270715536725361704207905, 2.21764130264741833360447193266, 2.69618901761837963349336940902, 4.12493522496975688811170544458, 4.37457729353707783153118195904, 5.21804425958131102468042130234, 5.88697573500241839839824341039, 7.17358589272389864933382324193, 8.07852437974061430205730312812, 8.55908794575148150718763387784, 9.04669255468697451499900986880, 9.95207913459940027223918929132, 10.66946457477229924609350294708, 11.247673509486865289746350411959, 12.12105556694469069147692498398, 12.41336833806140868418903279690, 13.31975689814465980727096923566, 13.95526211697680681541297864882, 14.47091531774550952409738871981, 15.45094802085605731269161190459, 16.58121024508826097778300084734, 16.845672528575886817762483739508, 17.54839036631374192025685996944, 18.15085832289638353452041527160

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