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Properties

Label 1-4021-4021.30-r1-0-0
Degree $1$
Conductor $4021$
Sign $0.740 - 0.672i$
Analytic cond. $432.116$
Root an. cond. $432.116$
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  + (0.697 − 0.716i)2-s + (−0.364 − 0.931i)3-s + (−0.0265 − 0.999i)4-s + (−0.778 + 0.627i)5-s + (−0.921 − 0.388i)6-s + (−0.587 + 0.809i)7-s + (−0.734 − 0.678i)8-s + (−0.733 + 0.679i)9-s + (−0.0936 + 0.995i)10-s + (0.980 − 0.197i)11-s + (−0.921 + 0.389i)12-s + (0.209 + 0.977i)13-s + (0.169 + 0.985i)14-s + (0.868 + 0.495i)15-s + (−0.998 + 0.0531i)16-s + (0.191 − 0.981i)17-s + ⋯
L(s)  = 1  + (0.697 − 0.716i)2-s + (−0.364 − 0.931i)3-s + (−0.0265 − 0.999i)4-s + (−0.778 + 0.627i)5-s + (−0.921 − 0.388i)6-s + (−0.587 + 0.809i)7-s + (−0.734 − 0.678i)8-s + (−0.733 + 0.679i)9-s + (−0.0936 + 0.995i)10-s + (0.980 − 0.197i)11-s + (−0.921 + 0.389i)12-s + (0.209 + 0.977i)13-s + (0.169 + 0.985i)14-s + (0.868 + 0.495i)15-s + (−0.998 + 0.0531i)16-s + (0.191 − 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4021\)
Sign: $0.740 - 0.672i$
Analytic conductor: \(432.116\)
Root analytic conductor: \(432.116\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4021} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4021,\ (1:\ ),\ 0.740 - 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.171258346 - 0.4524577791i\)
\(L(\frac12)\) \(\approx\) \(1.171258346 - 0.4524577791i\)
\(L(1)\) \(\approx\) \(0.8296182764 - 0.5183905781i\)
\(L(1)\) \(\approx\) \(0.8296182764 - 0.5183905781i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4021 \( 1 \)
good2 \( 1 + (0.697 - 0.716i)T \)
3 \( 1 + (-0.364 - 0.931i)T \)
5 \( 1 + (-0.778 + 0.627i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (0.980 - 0.197i)T \)
13 \( 1 + (0.209 + 0.977i)T \)
17 \( 1 + (0.191 - 0.981i)T \)
19 \( 1 + (0.829 + 0.558i)T \)
23 \( 1 + (-0.763 - 0.645i)T \)
29 \( 1 + (-0.873 + 0.486i)T \)
31 \( 1 + (0.808 - 0.589i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (-0.738 - 0.673i)T \)
43 \( 1 + (0.00625 - 0.999i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.998 - 0.0624i)T \)
59 \( 1 + (-0.611 + 0.791i)T \)
61 \( 1 + (0.638 - 0.769i)T \)
67 \( 1 + (-0.472 - 0.881i)T \)
71 \( 1 + (-0.335 + 0.941i)T \)
73 \( 1 + (-0.291 - 0.956i)T \)
79 \( 1 + (0.600 + 0.799i)T \)
83 \( 1 + (0.215 - 0.976i)T \)
89 \( 1 + (0.197 + 0.980i)T \)
97 \( 1 + (-0.821 - 0.569i)T \)
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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

−17.92974112848847425808555379610, −17.30740846454675818531651736204, −16.920788201621549191575655273389, −16.13116762297222875174406581400, −15.79317203633675559448822396222, −15.115953718999081388863300383467, −14.51668152750272125350931249983, −13.62212453250664050219913476574, −12.98296538154190719429511214546, −12.21682102011219560381020162942, −11.65602455846799932784678551433, −10.97644867002084046023811106417, −9.980851481042977029004460072752, −9.378597979185671266031737301426, −8.48551858093188558529720458681, −7.90350735018958904894907249913, −7.05633796969638331210524802948, −6.274385073642124521620173674689, −5.5986472783955426785933663144, −4.77636184569839264625815093559, −4.20454111488508013020321301282, −3.42697668388716153922994339926, −3.27271001419299284522215801328, −1.36726322428488081676241523084, −0.28079980165281091757848429460, 0.492172895527900391400456950171, 1.56907862141966002789613790949, 2.24755716528810777235186718940, 3.15770670318842887091603784673, 3.65386479941318495077258319747, 4.67254212648562014214130759391, 5.539553472862434829185393263528, 6.35147414345893657655103698289, 6.68620700377563912301262839844, 7.49562165932145261649345381578, 8.5608249430022112530958695830, 9.27084092610266067047017041807, 10.07638976145575554504734855357, 11.03864923421190465454976340466, 11.67249011385627117563557362742, 12.012580222230272085369697274494, 12.39577556686678991546721398104, 13.48652529228253585068786448327, 14.0774981032274960538643056471, 14.4708003139197863181005980092, 15.44040730467443187411888293420, 16.121951814573379154718008541452, 16.7262350295989159302866979480, 17.96409028056569833111232937096, 18.55398737033210496693067993464

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