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Properties

Label 1-4029-4029.1226-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.773 - 0.633i$
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  + (−0.822 − 0.568i)2-s + (0.354 + 0.935i)4-s + (−0.998 + 0.0603i)5-s + (−0.855 − 0.517i)7-s + (0.239 − 0.970i)8-s + (0.855 + 0.517i)10-s + (0.998 + 0.0603i)11-s + (0.354 + 0.935i)13-s + (0.410 + 0.911i)14-s + (−0.748 + 0.663i)16-s + (0.992 + 0.120i)19-s + (−0.410 − 0.911i)20-s + (−0.787 − 0.616i)22-s + (−0.707 − 0.707i)23-s + (0.992 − 0.120i)25-s + (0.239 − 0.970i)26-s + ⋯
L(s)  = 1  + (−0.822 − 0.568i)2-s + (0.354 + 0.935i)4-s + (−0.998 + 0.0603i)5-s + (−0.855 − 0.517i)7-s + (0.239 − 0.970i)8-s + (0.855 + 0.517i)10-s + (0.998 + 0.0603i)11-s + (0.354 + 0.935i)13-s + (0.410 + 0.911i)14-s + (−0.748 + 0.663i)16-s + (0.992 + 0.120i)19-s + (−0.410 − 0.911i)20-s + (−0.787 − 0.616i)22-s + (−0.707 − 0.707i)23-s + (0.992 − 0.120i)25-s + (0.239 − 0.970i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.773 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7334839451 - 0.2618705345i\)
\(L(\frac12)\) \(\approx\) \(0.7334839451 - 0.2618705345i\)
\(L(1)\) \(\approx\) \(0.5961417870 - 0.1349226032i\)
\(L(1)\) \(\approx\) \(0.5961417870 - 0.1349226032i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.822 - 0.568i)T \)
5 \( 1 + (-0.998 + 0.0603i)T \)
7 \( 1 + (-0.855 - 0.517i)T \)
11 \( 1 + (0.998 + 0.0603i)T \)
13 \( 1 + (0.354 + 0.935i)T \)
19 \( 1 + (0.992 + 0.120i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.954 + 0.297i)T \)
31 \( 1 + (-0.180 - 0.983i)T \)
37 \( 1 + (0.616 + 0.787i)T \)
41 \( 1 + (0.0603 + 0.998i)T \)
43 \( 1 + (0.663 - 0.748i)T \)
47 \( 1 + (-0.120 - 0.992i)T \)
53 \( 1 + (0.239 - 0.970i)T \)
59 \( 1 + (0.935 + 0.354i)T \)
61 \( 1 + (0.787 - 0.616i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (-0.855 + 0.517i)T \)
73 \( 1 + (-0.410 - 0.911i)T \)
83 \( 1 + (-0.935 + 0.354i)T \)
89 \( 1 + (-0.970 + 0.239i)T \)
97 \( 1 + (0.616 + 0.787i)T \)
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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

−18.56984180292947028603353996731, −17.85310565027168278601185032719, −17.22278090352970729638364027321, −16.213532058032802133730877815471, −16.011605960107397596567677317884, −15.411473573989817810736410018683, −14.688344492365129389706553285214, −14.01706794292901197796871654755, −13.0054436857797507796200856569, −12.24873850662209981616413687893, −11.560629742146454322980273511328, −10.9793822791641594560522898792, −10.07545378236283621666423871185, −9.33388099308125665591600596457, −8.89302649131594715457427204623, −8.03736555947531878224954129423, −7.41869932442123718972678665818, −6.805300010571752072157586138688, −5.845912941091872954162368066748, −5.484071311590620823216559780095, −4.233459061394522267999953353524, −3.46533862282195496771167957793, −2.68402771653092139410185054250, −1.44274878813577491904938356936, −0.60141855151330598426443169859, 0.55207339358935880264303477904, 1.38446173441800034632578228918, 2.42942306912868784365354995927, 3.4276281304185805811355773208, 3.88315963686805893791310922528, 4.42363389276147699179191591975, 5.93328351496660350960501059961, 6.89778857133934381421151601525, 7.09094932287481539638566306789, 8.07234299359214178224787877255, 8.70622101311770233558456148758, 9.53848856565137007546199369757, 9.91885276594197127918008218774, 10.91824305688685280401945567524, 11.58758352894984517337192892863, 11.88785721326893459761154611111, 12.77329308429242476626084862281, 13.423551511628282337758090071817, 14.32251776017369031666633416565, 15.11868275906489761455309183180, 16.09742783434226157406048416682, 16.42798164996313707186446946586, 16.85926181452985192171044288868, 17.84234838952902334289475844812, 18.71281741040106346602548185388

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