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Properties

Label 1-4024-4024.197-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.500 - 0.865i$
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  + (−0.780 + 0.625i)3-s + (−0.992 + 0.124i)5-s + (−0.229 + 0.973i)7-s + (0.217 − 0.976i)9-s + (−0.517 − 0.855i)11-s + (0.982 + 0.186i)13-s + (0.695 − 0.718i)15-s + (−0.925 + 0.378i)17-s + (0.180 − 0.983i)19-s + (−0.429 − 0.902i)21-s + (0.0187 − 0.999i)23-s + (0.968 − 0.247i)25-s + (0.441 + 0.897i)27-s + (0.993 + 0.112i)29-s + (−0.739 + 0.673i)31-s + ⋯
L(s)  = 1  + (−0.780 + 0.625i)3-s + (−0.992 + 0.124i)5-s + (−0.229 + 0.973i)7-s + (0.217 − 0.976i)9-s + (−0.517 − 0.855i)11-s + (0.982 + 0.186i)13-s + (0.695 − 0.718i)15-s + (−0.925 + 0.378i)17-s + (0.180 − 0.983i)19-s + (−0.429 − 0.902i)21-s + (0.0187 − 0.999i)23-s + (0.968 − 0.247i)25-s + (0.441 + 0.897i)27-s + (0.993 + 0.112i)29-s + (−0.739 + 0.673i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.500 - 0.865i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.500 - 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3776168732 - 0.2178759888i\)
\(L(\frac12)\) \(\approx\) \(0.3776168732 - 0.2178759888i\)
\(L(1)\) \(\approx\) \(0.5979343611 + 0.1253674884i\)
\(L(1)\) \(\approx\) \(0.5979343611 + 0.1253674884i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.780 + 0.625i)T \)
5 \( 1 + (-0.992 + 0.124i)T \)
7 \( 1 + (-0.229 + 0.973i)T \)
11 \( 1 + (-0.517 - 0.855i)T \)
13 \( 1 + (0.982 + 0.186i)T \)
17 \( 1 + (-0.925 + 0.378i)T \)
19 \( 1 + (0.180 - 0.983i)T \)
23 \( 1 + (0.0187 - 0.999i)T \)
29 \( 1 + (0.993 + 0.112i)T \)
31 \( 1 + (-0.739 + 0.673i)T \)
37 \( 1 + (0.772 + 0.635i)T \)
41 \( 1 + (-0.418 + 0.908i)T \)
43 \( 1 + (0.395 + 0.918i)T \)
47 \( 1 + (0.810 + 0.585i)T \)
53 \( 1 + (0.180 + 0.983i)T \)
59 \( 1 + (0.958 - 0.283i)T \)
61 \( 1 + (-0.939 + 0.343i)T \)
67 \( 1 + (-0.695 - 0.718i)T \)
71 \( 1 + (-0.871 - 0.490i)T \)
73 \( 1 + (-0.630 - 0.776i)T \)
79 \( 1 + (-0.590 - 0.806i)T \)
83 \( 1 + (-0.764 - 0.644i)T \)
89 \( 1 + (-0.996 - 0.0875i)T \)
97 \( 1 + (0.192 - 0.981i)T \)
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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

−18.54726907107113788531408289846, −17.8798448596335956889203492630, −17.3124462903516151226482394566, −16.50411560499067452982349567864, −15.92239159365537402887030663448, −15.49357987275216426429794003895, −14.443040042192968863931296413300, −13.50813693186062885812821700696, −13.14987440842274468197394331658, −12.37545188675475863986852816963, −11.74417784149905713507679424466, −11.04322512700792763064204879436, −10.54163954010737821811843716009, −9.7853391629159725113128698690, −8.63821765530413902708511025093, −7.92795794309028592903319382214, −7.21042570117843658542821811488, −6.97049653951015080069918011209, −5.87106969234535936423668805664, −5.19801425438442928832384550299, −4.15476037449206913218944037390, −3.89483359808843090817698726487, −2.63936064048301224995407310907, −1.5962433014663995497815994123, −0.7689917315409003771550463227, 0.20460884563552139026723114622, 1.27509184283166298879310334199, 2.85901887782544285117661911787, 3.12527445650192220399534198748, 4.364202797618392028281427708483, 4.63754615763900339392162743138, 5.73842540049080475286901061358, 6.279682208243685421032498517378, 6.91100853769836604211926231310, 8.072637214274087219506203498123, 8.78541098048182315018133599054, 9.09920537867098285319631197951, 10.35321580502504770174291756036, 10.92582701565265722141186929426, 11.368832298261291374376039134915, 12.02578670040819010927804616501, 12.75885816609827564191512012890, 13.4199151359746504515294917424, 14.550969028596859147719532080502, 15.22009360195563899810804563375, 15.794274204956058794985165441835, 16.12146495538679960844580509388, 16.74490624535196099204037270007, 17.92328110075482115019193092364, 18.23074017688450368105187660685

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