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Properties

Label 1-4024-4024.429-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.376 - 0.926i$
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  + (−0.920 − 0.389i)3-s + (0.630 + 0.776i)5-s + (0.517 + 0.855i)7-s + (0.695 + 0.718i)9-s + (−0.930 + 0.366i)11-s + (0.971 + 0.235i)13-s + (−0.277 − 0.960i)15-s + (0.764 − 0.644i)17-s + (−0.992 − 0.124i)19-s + (−0.143 − 0.989i)21-s + (−0.943 − 0.331i)23-s + (−0.205 + 0.978i)25-s + (−0.360 − 0.932i)27-s + (0.441 − 0.897i)29-s + (0.747 − 0.663i)31-s + ⋯
L(s)  = 1  + (−0.920 − 0.389i)3-s + (0.630 + 0.776i)5-s + (0.517 + 0.855i)7-s + (0.695 + 0.718i)9-s + (−0.930 + 0.366i)11-s + (0.971 + 0.235i)13-s + (−0.277 − 0.960i)15-s + (0.764 − 0.644i)17-s + (−0.992 − 0.124i)19-s + (−0.143 − 0.989i)21-s + (−0.943 − 0.331i)23-s + (−0.205 + 0.978i)25-s + (−0.360 − 0.932i)27-s + (0.441 − 0.897i)29-s + (0.747 − 0.663i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.376 - 0.926i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.376 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2872219670 - 0.4265911272i\)
\(L(\frac12)\) \(\approx\) \(0.2872219670 - 0.4265911272i\)
\(L(1)\) \(\approx\) \(0.7995133502 + 0.03242321984i\)
\(L(1)\) \(\approx\) \(0.7995133502 + 0.03242321984i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.920 - 0.389i)T \)
5 \( 1 + (0.630 + 0.776i)T \)
7 \( 1 + (0.517 + 0.855i)T \)
11 \( 1 + (-0.930 + 0.366i)T \)
13 \( 1 + (0.971 + 0.235i)T \)
17 \( 1 + (0.764 - 0.644i)T \)
19 \( 1 + (-0.992 - 0.124i)T \)
23 \( 1 + (-0.943 - 0.331i)T \)
29 \( 1 + (0.441 - 0.897i)T \)
31 \( 1 + (0.747 - 0.663i)T \)
37 \( 1 + (-0.984 + 0.174i)T \)
41 \( 1 + (-0.0812 - 0.996i)T \)
43 \( 1 + (0.507 - 0.861i)T \)
47 \( 1 + (0.265 - 0.964i)T \)
53 \( 1 + (-0.992 + 0.124i)T \)
59 \( 1 + (-0.452 - 0.891i)T \)
61 \( 1 + (-0.999 + 0.0250i)T \)
67 \( 1 + (0.277 - 0.960i)T \)
71 \( 1 + (-0.982 + 0.186i)T \)
73 \( 1 + (-0.958 - 0.283i)T \)
79 \( 1 + (-0.372 - 0.928i)T \)
83 \( 1 + (-0.998 - 0.0500i)T \)
89 \( 1 + (-0.00625 + 0.999i)T \)
97 \( 1 + (0.939 + 0.343i)T \)
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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

−18.399729891693053204898896643907, −17.79935688713303118457198885640, −17.34859454285768633469332924045, −16.66752814134130339567335665240, −16.07779047623664517193591188246, −15.620630666355440626125072112663, −14.487334474666454196430406921924, −13.90390876131638318297502109560, −13.01044201099428115958102518270, −12.68043330786360411053696956368, −11.78190781127277623135625444697, −10.93643044127425919378807997871, −10.35646425359361901866060842775, −10.08982227959737055070057027089, −8.92742132819983000412782182068, −8.26644590052572895166535557045, −7.60145895992588659869276224124, −6.41272718781753601526670332646, −5.966117704485892012226847251583, −5.24165489953655953952254080330, −4.54902862424734898517927484032, −3.93370178028115771185925064994, −2.93251317655003975853752716045, −1.39447587115234129396989116022, −1.26824882801341609279465616686, 0.16111563117266406928308354487, 1.60432969900706658209361095391, 2.14510387536367492268784093630, 2.87972558728215555413245654365, 4.12393702513234689351213408184, 4.97729868748695238900673983294, 5.72059430468838489622354599467, 6.125168846126059562746554448389, 6.88184179205086349766141628896, 7.736862870614455085247493708072, 8.34942984298792798090286029644, 9.36030566312783960852723037090, 10.283975918516829406659380721309, 10.59946434977026741551917644356, 11.420815781963255489326504435396, 12.05369213595185215406817111282, 12.636187294761205379872918470957, 13.66799982471266455308563667708, 13.87676647996045010261302166028, 15.00840692745727785590717067150, 15.58417736150465963127771840482, 16.15123094204065414506763410698, 17.24296418234222715469274444589, 17.571859156137635806462110533575, 18.31273308790333484390084660542

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