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Properties

Label 1-4024-4024.1821-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.809 + 0.586i$
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  + (0.485 − 0.874i)3-s + (−0.0937 + 0.995i)5-s + (0.217 − 0.976i)7-s + (−0.528 − 0.848i)9-s + (−0.695 − 0.718i)11-s + (−0.600 + 0.799i)13-s + (0.824 + 0.565i)15-s + (0.474 + 0.880i)17-s + (0.253 − 0.967i)19-s + (−0.747 − 0.663i)21-s + (−0.395 − 0.918i)23-s + (−0.982 − 0.186i)25-s + (−0.998 + 0.0500i)27-s + (−0.764 − 0.644i)29-s + (−0.229 − 0.973i)31-s + ⋯
L(s)  = 1  + (0.485 − 0.874i)3-s + (−0.0937 + 0.995i)5-s + (0.217 − 0.976i)7-s + (−0.528 − 0.848i)9-s + (−0.695 − 0.718i)11-s + (−0.600 + 0.799i)13-s + (0.824 + 0.565i)15-s + (0.474 + 0.880i)17-s + (0.253 − 0.967i)19-s + (−0.747 − 0.663i)21-s + (−0.395 − 0.918i)23-s + (−0.982 − 0.186i)25-s + (−0.998 + 0.0500i)27-s + (−0.764 − 0.644i)29-s + (−0.229 − 0.973i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.809 + 0.586i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.809 + 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1077482426 - 0.3323015875i\)
\(L(\frac12)\) \(\approx\) \(-0.1077482426 - 0.3323015875i\)
\(L(1)\) \(\approx\) \(0.8912598740 - 0.3155033262i\)
\(L(1)\) \(\approx\) \(0.8912598740 - 0.3155033262i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.485 - 0.874i)T \)
5 \( 1 + (-0.0937 + 0.995i)T \)
7 \( 1 + (0.217 - 0.976i)T \)
11 \( 1 + (-0.695 - 0.718i)T \)
13 \( 1 + (-0.600 + 0.799i)T \)
17 \( 1 + (0.474 + 0.880i)T \)
19 \( 1 + (0.253 - 0.967i)T \)
23 \( 1 + (-0.395 - 0.918i)T \)
29 \( 1 + (-0.764 - 0.644i)T \)
31 \( 1 + (-0.229 - 0.973i)T \)
37 \( 1 + (-0.265 + 0.964i)T \)
41 \( 1 + (0.0687 - 0.997i)T \)
43 \( 1 + (0.996 - 0.0875i)T \)
47 \( 1 + (0.452 + 0.891i)T \)
53 \( 1 + (0.253 + 0.967i)T \)
59 \( 1 + (-0.539 - 0.842i)T \)
61 \( 1 + (0.965 + 0.259i)T \)
67 \( 1 + (-0.824 + 0.565i)T \)
71 \( 1 + (0.920 + 0.389i)T \)
73 \( 1 + (0.118 - 0.992i)T \)
79 \( 1 + (-0.0812 + 0.996i)T \)
83 \( 1 + (-0.864 + 0.501i)T \)
89 \( 1 + (0.659 - 0.752i)T \)
97 \( 1 + (-0.858 + 0.512i)T \)
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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

−18.962499069950936570809224882924, −18.05905974707899768128728324087, −17.59684097729078922198394964983, −16.55722661946858846168503657603, −16.11214992888358185930474089611, −15.53662024995937769885662977153, −14.90459564798151399023721098091, −14.297871290961620392848901755, −13.4187582104419715833450540435, −12.58338800758285657164500636688, −12.182401889805915320952289377677, −11.35965677945185600270510125178, −10.41189314346189215473977520156, −9.70560106877277867996242979880, −9.30812393676407661103055865767, −8.50316721721676648649288390855, −7.85114442459380186697858506379, −7.34717695084636162881388113634, −5.62907739296093640147410224877, −5.402695262123432090438437560648, −4.861946721841160688218191087232, −3.903577005664857518232168652103, −3.09529228816058279532865332906, −2.29617991338605931138525309478, −1.50132390447980882769611715586, 0.08813089468340109886450990483, 1.148790557282005947210995657041, 2.25972280341274424108460198249, 2.684819680482349490417216479829, 3.69268456304111709595894551398, 4.23068915827766199683289875417, 5.51115020772755963644619256933, 6.32519344565624351242994461645, 6.89337273734759855612730546355, 7.67283585560711116145343341097, 7.92782720290854851783867912211, 8.96063747277052777835712209046, 9.78696286021615190954326379209, 10.604920410965026262679728237940, 11.147289059853290246615115700203, 11.85261286113274334375247052345, 12.70848120815718775395265517577, 13.487094428502977889552399482160, 13.936767189770080306903604256009, 14.50207551426802231163004099643, 15.131294859355288913115782588830, 15.96190473845358261667131304280, 17.05441465582738418399465717226, 17.30003888778418186867040531010, 18.249025172416310778337080415767

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