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Properties

Label 1-4024-4024.117-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.886 - 0.462i$
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  + (0.998 − 0.0625i)3-s + (−0.677 + 0.735i)5-s + (−0.349 − 0.937i)7-s + (0.992 − 0.124i)9-s + (0.990 − 0.137i)11-s + (0.325 + 0.945i)13-s + (−0.630 + 0.776i)15-s + (0.780 − 0.625i)17-s + (−0.539 + 0.842i)19-s + (−0.407 − 0.913i)21-s + (−0.905 − 0.424i)23-s + (−0.0812 − 0.996i)25-s + (0.982 − 0.186i)27-s + (0.871 − 0.490i)29-s + (0.889 − 0.457i)31-s + ⋯
L(s)  = 1  + (0.998 − 0.0625i)3-s + (−0.677 + 0.735i)5-s + (−0.349 − 0.937i)7-s + (0.992 − 0.124i)9-s + (0.990 − 0.137i)11-s + (0.325 + 0.945i)13-s + (−0.630 + 0.776i)15-s + (0.780 − 0.625i)17-s + (−0.539 + 0.842i)19-s + (−0.407 − 0.913i)21-s + (−0.905 − 0.424i)23-s + (−0.0812 − 0.996i)25-s + (0.982 − 0.186i)27-s + (0.871 − 0.490i)29-s + (0.889 − 0.457i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.886 - 0.462i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.390457474 - 0.5854113729i\)
\(L(\frac12)\) \(\approx\) \(2.390457474 - 0.5854113729i\)
\(L(1)\) \(\approx\) \(1.446468233 - 0.07702314788i\)
\(L(1)\) \(\approx\) \(1.446468233 - 0.07702314788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.998 - 0.0625i)T \)
5 \( 1 + (-0.677 + 0.735i)T \)
7 \( 1 + (-0.349 - 0.937i)T \)
11 \( 1 + (0.990 - 0.137i)T \)
13 \( 1 + (0.325 + 0.945i)T \)
17 \( 1 + (0.780 - 0.625i)T \)
19 \( 1 + (-0.539 + 0.842i)T \)
23 \( 1 + (-0.905 - 0.424i)T \)
29 \( 1 + (0.871 - 0.490i)T \)
31 \( 1 + (0.889 - 0.457i)T \)
37 \( 1 + (-0.168 - 0.985i)T \)
41 \( 1 + (0.795 + 0.605i)T \)
43 \( 1 + (0.441 - 0.897i)T \)
47 \( 1 + (-0.549 - 0.835i)T \)
53 \( 1 + (-0.539 - 0.842i)T \)
59 \( 1 + (0.817 + 0.575i)T \)
61 \( 1 + (-0.980 - 0.198i)T \)
67 \( 1 + (0.630 + 0.776i)T \)
71 \( 1 + (0.0687 + 0.997i)T \)
73 \( 1 + (-0.668 - 0.743i)T \)
79 \( 1 + (-0.996 + 0.0875i)T \)
83 \( 1 + (-0.920 + 0.389i)T \)
89 \( 1 + (0.998 - 0.0500i)T \)
97 \( 1 + (-0.943 - 0.331i)T \)
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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

−18.830480545515228207774771304508, −17.816470455346271312695043817844, −17.23320784400326830826349348557, −16.14515704959336635763820256322, −15.773028054877094605431133041437, −15.1847776068749639845815613503, −14.563590659957622498903709634268, −13.78631737678762615561523872691, −12.91136211853977243650566343604, −12.44910889903935401332554586277, −11.93697195422301412548606113907, −10.95014415391981670793917651883, −9.98992498150711113737552325088, −9.38585242258290369699253813081, −8.66443809976495113424462770939, −8.26655654204934094568934946247, −7.60605943916194371124240821535, −6.57682417353742510704846615178, −5.875079621203162641375072496534, −4.83055282474842992774793479462, −4.22432719228052286737133190952, −3.32709282327041025289075380819, −2.87116152056840794522311026671, −1.70526221598737420566277861649, −1.00032187622479033521643790939, 0.70914941934193765499714473907, 1.70274506292181457785885146444, 2.62733676845353522344910525642, 3.45294826439756098868444602909, 4.118605768375418515958548202422, 4.31449490067024557089659793091, 6.02078159524385415751712573243, 6.71148806365870632343751372531, 7.18231118039084058950900507974, 8.001081540547862191271494973471, 8.50735046865297464645742260137, 9.52964018602925578389188295594, 10.026458491381567970740741376161, 10.706903473054591553627724284786, 11.70438058782818172377270979438, 12.12484039553555766378551141173, 13.11037158828974558953846253678, 14.00568753297063872444323912497, 14.25242817043353097898324347273, 14.71866125623301548979529636077, 15.8361486667484463832547642776, 16.18466829887503118023570374189, 16.933094692848008341229367005671, 17.88943258859112930448908412602, 18.729684034978699652383783793398

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