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Properties

Label 1-4024-4024.1269-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.864 - 0.502i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (−0.731 + 0.682i)3-s + (0.883 − 0.468i)5-s + (−0.418 + 0.908i)7-s + (0.0687 − 0.997i)9-s + (0.0812 + 0.996i)11-s + (0.668 − 0.743i)13-s + (−0.325 + 0.945i)15-s + (−0.253 − 0.967i)17-s + (−0.852 − 0.523i)19-s + (−0.313 − 0.949i)21-s + (0.517 − 0.855i)23-s + (0.560 − 0.828i)25-s + (0.630 + 0.776i)27-s + (−0.992 − 0.124i)29-s + (0.838 + 0.544i)31-s + ⋯
L(s)  = 1  + (−0.731 + 0.682i)3-s + (0.883 − 0.468i)5-s + (−0.418 + 0.908i)7-s + (0.0687 − 0.997i)9-s + (0.0812 + 0.996i)11-s + (0.668 − 0.743i)13-s + (−0.325 + 0.945i)15-s + (−0.253 − 0.967i)17-s + (−0.852 − 0.523i)19-s + (−0.313 − 0.949i)21-s + (0.517 − 0.855i)23-s + (0.560 − 0.828i)25-s + (0.630 + 0.776i)27-s + (−0.992 − 0.124i)29-s + (0.838 + 0.544i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.864 - 0.502i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.864 - 0.502i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.231550581 - 0.3319824887i\)
\(L(\frac12)\) \(\approx\) \(1.231550581 - 0.3319824887i\)
\(L(1)\) \(\approx\) \(0.9284972588 + 0.08550407242i\)
\(L(1)\) \(\approx\) \(0.9284972588 + 0.08550407242i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.731 + 0.682i)T \)
5 \( 1 + (0.883 - 0.468i)T \)
7 \( 1 + (-0.418 + 0.908i)T \)
11 \( 1 + (0.0812 + 0.996i)T \)
13 \( 1 + (0.668 - 0.743i)T \)
17 \( 1 + (-0.253 - 0.967i)T \)
19 \( 1 + (-0.852 - 0.523i)T \)
23 \( 1 + (0.517 - 0.855i)T \)
29 \( 1 + (-0.992 - 0.124i)T \)
31 \( 1 + (0.838 + 0.544i)T \)
37 \( 1 + (0.441 + 0.897i)T \)
41 \( 1 + (0.0437 + 0.999i)T \)
43 \( 1 + (-0.695 + 0.718i)T \)
47 \( 1 + (0.764 - 0.644i)T \)
53 \( 1 + (-0.852 + 0.523i)T \)
59 \( 1 + (-0.474 - 0.880i)T \)
61 \( 1 + (0.739 - 0.673i)T \)
67 \( 1 + (0.325 + 0.945i)T \)
71 \( 1 + (0.677 - 0.735i)T \)
73 \( 1 + (-0.803 - 0.595i)T \)
79 \( 1 + (0.496 - 0.868i)T \)
83 \( 1 + (-0.0937 - 0.995i)T \)
89 \( 1 + (0.824 - 0.565i)T \)
97 \( 1 + (-0.610 - 0.791i)T \)
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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

−18.69293829228257062524823943421, −17.706101753102861075067543967270, −17.109028171724402897482385099815, −16.7970529805922857142237757930, −16.07568095602997275444551117394, −15.08569923722024482412320303038, −14.0987660928524305575319878186, −13.7024689036444767343597731772, −13.140690235031472281513156064744, −12.566602915466174316298261311096, −11.44353387093974475352067569791, −10.90685425500385915733696435288, −10.540680300466040278388502791216, −9.63064350156339295231034945101, −8.79714536851133223922826108264, −7.94577477468690562223342691625, −7.08443495829885806827367282994, −6.499198685515414774638068100709, −5.98294076203555868370531675515, −5.41502941273718799591148668528, −4.13686360092968263368573306336, −3.594338785624151077688423357869, −2.41241478915490848833016509272, −1.64408133198970432024960255146, −0.91562230753857966416145310639, 0.46374157210865578508849272465, 1.57647498071186018951629248021, 2.55916178217140677222482094044, 3.24020713167136830829090088712, 4.59826640479563202391706003447, 4.81578795884010269731249150600, 5.67095547678506069327412924173, 6.36761008221974195930941062630, 6.799135614183424585371305291249, 8.17716187533250140537347186967, 8.94749671675396079233521944719, 9.47327340941509634865422655827, 10.03413215383815732817720495921, 10.759496783743094392939807690720, 11.54962149953496957782986595948, 12.29175047240937526373515992178, 12.905916377239389794013366466560, 13.37150392914695620176823434326, 14.60022257461081584500864962388, 15.117884133424481117998760004593, 15.75442988018576266529881483158, 16.36761152664015078053008700645, 17.126510638091974603775478215636, 17.63596421240563835912956117627, 18.27513613732957429708502346540

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