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Properties

Label 1-4024-4024.69-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.599 + 0.800i$
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  + (−0.677 − 0.735i)3-s + (−0.659 − 0.752i)5-s + (−0.590 − 0.806i)7-s + (−0.0812 + 0.996i)9-s + (0.372 + 0.928i)11-s + (−0.289 + 0.957i)13-s + (−0.106 + 0.994i)15-s + (0.992 + 0.124i)17-s + (−0.795 − 0.605i)19-s + (−0.192 + 0.981i)21-s + (0.877 + 0.479i)23-s + (−0.131 + 0.991i)25-s + (0.787 − 0.615i)27-s + (0.990 − 0.137i)29-s + (0.313 + 0.949i)31-s + ⋯
L(s)  = 1  + (−0.677 − 0.735i)3-s + (−0.659 − 0.752i)5-s + (−0.590 − 0.806i)7-s + (−0.0812 + 0.996i)9-s + (0.372 + 0.928i)11-s + (−0.289 + 0.957i)13-s + (−0.106 + 0.994i)15-s + (0.992 + 0.124i)17-s + (−0.795 − 0.605i)19-s + (−0.192 + 0.981i)21-s + (0.877 + 0.479i)23-s + (−0.131 + 0.991i)25-s + (0.787 − 0.615i)27-s + (0.990 − 0.137i)29-s + (0.313 + 0.949i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.599 + 0.800i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.599 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01351562046 + 0.02700022625i\)
\(L(\frac12)\) \(\approx\) \(0.01351562046 + 0.02700022625i\)
\(L(1)\) \(\approx\) \(0.6089674756 - 0.1852583293i\)
\(L(1)\) \(\approx\) \(0.6089674756 - 0.1852583293i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.677 - 0.735i)T \)
5 \( 1 + (-0.659 - 0.752i)T \)
7 \( 1 + (-0.590 - 0.806i)T \)
11 \( 1 + (0.372 + 0.928i)T \)
13 \( 1 + (-0.289 + 0.957i)T \)
17 \( 1 + (0.992 + 0.124i)T \)
19 \( 1 + (-0.795 - 0.605i)T \)
23 \( 1 + (0.877 + 0.479i)T \)
29 \( 1 + (0.990 - 0.137i)T \)
31 \( 1 + (0.313 + 0.949i)T \)
37 \( 1 + (0.0312 - 0.999i)T \)
41 \( 1 + (0.496 - 0.868i)T \)
43 \( 1 + (-0.930 + 0.366i)T \)
47 \( 1 + (-0.441 + 0.897i)T \)
53 \( 1 + (-0.795 + 0.605i)T \)
59 \( 1 + (-0.764 - 0.644i)T \)
61 \( 1 + (-0.429 - 0.902i)T \)
67 \( 1 + (0.106 + 0.994i)T \)
71 \( 1 + (0.560 - 0.828i)T \)
73 \( 1 + (-0.925 + 0.378i)T \)
79 \( 1 + (0.994 - 0.0999i)T \)
83 \( 1 + (0.630 + 0.776i)T \)
89 \( 1 + (-0.277 - 0.960i)T \)
97 \( 1 + (-0.998 + 0.0625i)T \)
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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

−18.38099839883569239473515499145, −17.500866118418243561729718261, −16.55223202029127106889905136503, −16.40662082397188799378178123008, −15.363921858870434038718095666322, −15.04015765959649681192008345526, −14.49972203544434933187358269835, −13.39889389442201739048092285688, −12.474043520897310369318266906093, −11.968398751892470885524787958207, −11.39796648497924889416480647395, −10.547969691516468899807644007934, −10.154857003472437669689720493143, −9.35035662139234421026146390042, −8.43114757467907934255671880762, −7.91543610590279194518083382175, −6.62509662145672529620244251771, −6.322736194519062990763313131978, −5.535392984112954610085496214526, −4.78529218462793414554167976889, −3.78869312861971708282833362759, −3.159850640315372534615589586615, −2.70102747981821517680347899059, −1.04344536822215158500095902685, −0.01196039253578193736993800417, 1.08966235851954593011534334434, 1.64048876058944352443335267389, 2.835820941003688916160727635315, 3.880428000071712148752205046, 4.62173467196051615832582548828, 5.07234809338727360061684437939, 6.26101387285954913195089788093, 6.85190996672449565240869072580, 7.39469726139975155939270054942, 8.05415161439964297940333928407, 9.04692810661170375358186373730, 9.67147233060922009956476537324, 10.60507721437991129954321419795, 11.23073423218922791856437427335, 12.08032960200863120025616951561, 12.493577076763702752318838637806, 12.98646136247466143449538376593, 13.87458930357024456840411836797, 14.46889364341238730925141219548, 15.55331044891190666661963659476, 16.13336847782930933691240353823, 16.903320407036363746206892887, 17.15585708300293691152674943692, 17.84210834063121822557777999000, 18.9459916352019589505762413054

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