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Properties

Label 1-4024-4024.189-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.135 - 0.990i$
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  + (−0.539 − 0.842i)3-s + (−0.795 − 0.605i)5-s + (0.838 − 0.544i)7-s + (−0.418 + 0.908i)9-s + (0.590 − 0.806i)11-s + (−0.560 + 0.828i)13-s + (−0.0812 + 0.996i)15-s + (−0.180 − 0.983i)17-s + (0.951 + 0.307i)19-s + (−0.910 − 0.412i)21-s + (0.747 − 0.663i)23-s + (0.265 + 0.964i)25-s + (0.990 − 0.137i)27-s + (0.349 − 0.937i)29-s + (0.241 + 0.970i)31-s + ⋯
L(s)  = 1  + (−0.539 − 0.842i)3-s + (−0.795 − 0.605i)5-s + (0.838 − 0.544i)7-s + (−0.418 + 0.908i)9-s + (0.590 − 0.806i)11-s + (−0.560 + 0.828i)13-s + (−0.0812 + 0.996i)15-s + (−0.180 − 0.983i)17-s + (0.951 + 0.307i)19-s + (−0.910 − 0.412i)21-s + (0.747 − 0.663i)23-s + (0.265 + 0.964i)25-s + (0.990 − 0.137i)27-s + (0.349 − 0.937i)29-s + (0.241 + 0.970i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.135 - 0.990i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.135 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.098739211 - 0.9584547623i\)
\(L(\frac12)\) \(\approx\) \(1.098739211 - 0.9584547623i\)
\(L(1)\) \(\approx\) \(0.8527894741 - 0.3998870362i\)
\(L(1)\) \(\approx\) \(0.8527894741 - 0.3998870362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.539 - 0.842i)T \)
5 \( 1 + (-0.795 - 0.605i)T \)
7 \( 1 + (0.838 - 0.544i)T \)
11 \( 1 + (0.590 - 0.806i)T \)
13 \( 1 + (-0.560 + 0.828i)T \)
17 \( 1 + (-0.180 - 0.983i)T \)
19 \( 1 + (0.951 + 0.307i)T \)
23 \( 1 + (0.747 - 0.663i)T \)
29 \( 1 + (0.349 - 0.937i)T \)
31 \( 1 + (0.241 + 0.970i)T \)
37 \( 1 + (0.905 - 0.424i)T \)
41 \( 1 + (-0.549 + 0.835i)T \)
43 \( 1 + (-0.517 + 0.855i)T \)
47 \( 1 + (-0.993 + 0.112i)T \)
53 \( 1 + (0.951 - 0.307i)T \)
59 \( 1 + (0.925 - 0.378i)T \)
61 \( 1 + (0.998 - 0.0625i)T \)
67 \( 1 + (0.0812 + 0.996i)T \)
71 \( 1 + (0.452 + 0.891i)T \)
73 \( 1 + (0.659 - 0.752i)T \)
79 \( 1 + (0.168 + 0.985i)T \)
83 \( 1 + (-0.992 - 0.124i)T \)
89 \( 1 + (0.695 + 0.718i)T \)
97 \( 1 + (0.640 + 0.768i)T \)
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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

−18.39670892811276522978777468703, −17.90117811492326464509845707606, −17.27763344916240694793096829664, −16.684138835504496506401165968197, −15.59176162322542376982541568707, −15.24376356407459994183053663274, −14.85992973385209482094399539138, −14.21624045326706678321420635544, −12.98453687533970352960590552659, −12.148271906948083882165275969812, −11.69049968001139352722830848118, −11.152297170934599398094908027310, −10.38383541584117368925674407215, −9.8055395610678608779041457948, −8.92329900352192207344571568664, −8.22319633351034707413327530482, −7.38452415593811733053505428, −6.7495961352729447447089818346, −5.75373946721038430848045586798, −5.077776987210410727380077675940, −4.47177551240847508936674581666, −3.644204183089376075999710860034, −2.982155023912878580519995348428, −1.93451045975214192734669976704, −0.74230745992443153252910342885, 0.78646312860370126583711936783, 1.08830144895621037955646415960, 2.19838179098894447534065744281, 3.20731515263005124685437998633, 4.24560446981283200562217642433, 4.87418331211716917780746564206, 5.417431025322215427160650319845, 6.65288801124949297702817261825, 6.99663447816263932175529248264, 7.90785880051421785711647764508, 8.30578080520969166506588701459, 9.151541694710947074095275143444, 10.07769859608996636855074220368, 11.20759156870057666397056409936, 11.59256094305858338442371247319, 11.82025975716538958215814750041, 12.84278585475819752396778614196, 13.482666241442169108795166565859, 14.22562337089335754572006007907, 14.651106780759191109748710718496, 15.89464272827415900995590388261, 16.52827336162616588902295268860, 16.793947979522608293091879326376, 17.66403268429780276222562121299, 18.2908374600640179995862544278

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