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Properties

Label 1-4024-4024.2131-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.990 - 0.134i$
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  + (0.764 − 0.644i)3-s + (0.695 + 0.718i)5-s + (−0.974 + 0.223i)7-s + (0.168 − 0.985i)9-s + (−0.610 + 0.791i)11-s + (−0.360 + 0.932i)13-s + (0.994 + 0.0999i)15-s + (−0.337 − 0.941i)17-s + (−0.217 − 0.976i)19-s + (−0.600 + 0.799i)21-s + (−0.192 + 0.981i)23-s + (−0.0312 + 0.999i)25-s + (−0.507 − 0.861i)27-s + (−0.395 − 0.918i)29-s + (0.955 − 0.295i)31-s + ⋯
L(s)  = 1  + (0.764 − 0.644i)3-s + (0.695 + 0.718i)5-s + (−0.974 + 0.223i)7-s + (0.168 − 0.985i)9-s + (−0.610 + 0.791i)11-s + (−0.360 + 0.932i)13-s + (0.994 + 0.0999i)15-s + (−0.337 − 0.941i)17-s + (−0.217 − 0.976i)19-s + (−0.600 + 0.799i)21-s + (−0.192 + 0.981i)23-s + (−0.0312 + 0.999i)25-s + (−0.507 − 0.861i)27-s + (−0.395 − 0.918i)29-s + (0.955 − 0.295i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.990 - 0.134i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (2131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.990 - 0.134i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01092433832 - 0.1621888933i\)
\(L(\frac12)\) \(\approx\) \(0.01092433832 - 0.1621888933i\)
\(L(1)\) \(\approx\) \(1.039644052 - 0.06024371940i\)
\(L(1)\) \(\approx\) \(1.039644052 - 0.06024371940i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.764 - 0.644i)T \)
5 \( 1 + (0.695 + 0.718i)T \)
7 \( 1 + (-0.974 + 0.223i)T \)
11 \( 1 + (-0.610 + 0.791i)T \)
13 \( 1 + (-0.360 + 0.932i)T \)
17 \( 1 + (-0.337 - 0.941i)T \)
19 \( 1 + (-0.217 - 0.976i)T \)
23 \( 1 + (-0.192 + 0.981i)T \)
29 \( 1 + (-0.395 - 0.918i)T \)
31 \( 1 + (0.955 - 0.295i)T \)
37 \( 1 + (-0.301 + 0.953i)T \)
41 \( 1 + (-0.968 + 0.247i)T \)
43 \( 1 + (-0.858 + 0.512i)T \)
47 \( 1 + (0.649 - 0.760i)T \)
53 \( 1 + (0.217 - 0.976i)T \)
59 \( 1 + (-0.349 - 0.937i)T \)
61 \( 1 + (-0.0437 + 0.999i)T \)
67 \( 1 + (0.994 - 0.0999i)T \)
71 \( 1 + (-0.441 - 0.897i)T \)
73 \( 1 + (-0.277 + 0.960i)T \)
79 \( 1 + (-0.962 - 0.271i)T \)
83 \( 1 + (-0.996 + 0.0875i)T \)
89 \( 1 + (0.372 - 0.928i)T \)
97 \( 1 + (-0.817 + 0.575i)T \)
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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

−18.9301848740245759622785549029, −18.22085432804936715441569587054, −17.103150155969747935367363915897, −16.775916763387491339838706281265, −16.01666527730373869540676033868, −15.55556433529586850951525364940, −14.68711758746313228546959545946, −13.94945784537923349808530873258, −13.40022509563850966403077612522, −12.70729232110889784022800750281, −12.34270350221087065845767690630, −10.76416521868740258698378406168, −10.38482031827846795846431289059, −9.92887007677249583380949716376, −9.00831219493391303683373688739, −8.48555998296026175688112390453, −7.94204301453510105662312770014, −6.859464426974365188817320031530, −5.88318679694511828592035888046, −5.45544006135112868690158709082, −4.49354145298799969995802184841, −3.731511967942893118932837734577, −2.98363005725885413517768649564, −2.31249864594837525071279490595, −1.280952520118217634974148242019, 0.03604246650391683481923905242, 1.54882886218873459914829500366, 2.37365669045549402734953718895, 2.71434736895041945033697431662, 3.55436296217738071040253108405, 4.58531592549145630633064287890, 5.52036170267799864985537171608, 6.56623406283686809219177145515, 6.81963477263420725378512949437, 7.439165524856418924711764026316, 8.41722252885287910518492263853, 9.358937375953936773566252774191, 9.67210716300666132488078695149, 10.2311135164572993032800192914, 11.5617078612970268661790257278, 11.87962437265201435639518340751, 13.034775999053366089506241965636, 13.40251657010569913127451918373, 13.81830943513595786991223410037, 14.770753120539298639551581507887, 15.35218293335359704598986391226, 15.83390850843281367344796154251, 17.06175760534623978299913583680, 17.51580643113773759580911252972, 18.437652749592778973010890823943

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