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Properties

Label 1-4024-4024.285-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.619 + 0.784i$
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  + (0.925 − 0.378i)3-s + (−0.217 + 0.976i)5-s + (−0.999 + 0.0125i)7-s + (0.713 − 0.700i)9-s + (−0.974 + 0.223i)11-s + (0.441 − 0.897i)13-s + (0.168 + 0.985i)15-s + (−0.106 − 0.994i)17-s + (−0.997 − 0.0750i)19-s + (−0.920 + 0.389i)21-s + (0.910 − 0.412i)23-s + (−0.905 − 0.424i)25-s + (0.395 − 0.918i)27-s + (0.831 + 0.554i)29-s + (−0.485 + 0.874i)31-s + ⋯
L(s)  = 1  + (0.925 − 0.378i)3-s + (−0.217 + 0.976i)5-s + (−0.999 + 0.0125i)7-s + (0.713 − 0.700i)9-s + (−0.974 + 0.223i)11-s + (0.441 − 0.897i)13-s + (0.168 + 0.985i)15-s + (−0.106 − 0.994i)17-s + (−0.997 − 0.0750i)19-s + (−0.920 + 0.389i)21-s + (0.910 − 0.412i)23-s + (−0.905 − 0.424i)25-s + (0.395 − 0.918i)27-s + (0.831 + 0.554i)29-s + (−0.485 + 0.874i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.619 + 0.784i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.619 + 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2856245454 + 0.5894782351i\)
\(L(\frac12)\) \(\approx\) \(0.2856245454 + 0.5894782351i\)
\(L(1)\) \(\approx\) \(1.010384676 + 0.03229692128i\)
\(L(1)\) \(\approx\) \(1.010384676 + 0.03229692128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.925 - 0.378i)T \)
5 \( 1 + (-0.217 + 0.976i)T \)
7 \( 1 + (-0.999 + 0.0125i)T \)
11 \( 1 + (-0.974 + 0.223i)T \)
13 \( 1 + (0.441 - 0.897i)T \)
17 \( 1 + (-0.106 - 0.994i)T \)
19 \( 1 + (-0.997 - 0.0750i)T \)
23 \( 1 + (0.910 - 0.412i)T \)
29 \( 1 + (0.831 + 0.554i)T \)
31 \( 1 + (-0.485 + 0.874i)T \)
37 \( 1 + (-0.407 - 0.913i)T \)
41 \( 1 + (-0.934 + 0.355i)T \)
43 \( 1 + (-0.313 + 0.949i)T \)
47 \( 1 + (0.889 + 0.457i)T \)
53 \( 1 + (-0.997 + 0.0750i)T \)
59 \( 1 + (0.277 + 0.960i)T \)
61 \( 1 + (0.817 + 0.575i)T \)
67 \( 1 + (-0.168 + 0.985i)T \)
71 \( 1 + (-0.993 + 0.112i)T \)
73 \( 1 + (0.695 + 0.718i)T \)
79 \( 1 + (-0.756 - 0.654i)T \)
83 \( 1 + (-0.337 + 0.941i)T \)
89 \( 1 + (-0.590 + 0.806i)T \)
97 \( 1 + (-0.668 - 0.743i)T \)
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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

−18.64330905844338762613047453656, −17.21961413750769356962289906928, −16.81192399127528059307711827027, −16.08835800592650032922654353861, −15.44270645225611356939941987859, −15.16211730315162858140456681036, −13.93334396572081744701100058483, −13.40373504618555386617738124675, −12.92879823159847326410631708227, −12.330186645340723271201038651912, −11.27349975807810745518398304225, −10.43817392244978177888038619178, −9.85856920806519137117007002598, −9.06058961881122012941022424525, −8.5639971487739115461511396740, −8.04113315751567058082952003292, −7.07553244515318989877735401429, −6.28186126508523500423908857627, −5.35984368619383740609500891366, −4.54427226650730470196933730979, −3.86736248755327914622077760374, −3.26453517550481668138448112147, −2.25068336773688252136275287714, −1.555350418980267009211234603605, −0.16002395511820221007167841710, 1.09541283864761967025264482407, 2.46101579015812414756196334052, 2.81618751145742134031581095063, 3.35233394184202628576071073491, 4.26601842332626893373403633461, 5.33213243012412380119699759568, 6.30564411172120823011606636433, 6.97781605487182544672144517040, 7.36429428755297187534976251447, 8.30882525898972846701500823918, 8.8697725688612755533147673580, 9.80117014431721473814026442864, 10.407751582493123191261665908392, 10.905879232994962168083262325134, 12.04228050915083054954870913415, 12.88186380365626545135991340244, 13.0932384011809251704509980111, 13.99791739598001171416090174722, 14.60140915135315538379936477947, 15.40381854553204120995598157968, 15.70658141698989845564409084839, 16.45998746998288491473629130892, 17.80678970109292336578553884389, 18.05436354614190278428014202866, 18.86933687696873371966438469675

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