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Properties

Label 1-4024-4024.237-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.999 + 0.0307i$
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  + (−0.600 − 0.799i)3-s + (0.958 + 0.283i)5-s + (0.930 − 0.366i)7-s + (−0.277 + 0.960i)9-s + (−0.889 + 0.457i)11-s + (0.418 + 0.908i)13-s + (−0.349 − 0.937i)15-s + (0.998 − 0.0500i)17-s + (0.630 − 0.776i)19-s + (−0.852 − 0.523i)21-s + (0.980 − 0.198i)23-s + (0.838 + 0.544i)25-s + (0.934 − 0.355i)27-s + (−0.360 + 0.932i)29-s + (0.877 − 0.479i)31-s + ⋯
L(s)  = 1  + (−0.600 − 0.799i)3-s + (0.958 + 0.283i)5-s + (0.930 − 0.366i)7-s + (−0.277 + 0.960i)9-s + (−0.889 + 0.457i)11-s + (0.418 + 0.908i)13-s + (−0.349 − 0.937i)15-s + (0.998 − 0.0500i)17-s + (0.630 − 0.776i)19-s + (−0.852 − 0.523i)21-s + (0.980 − 0.198i)23-s + (0.838 + 0.544i)25-s + (0.934 − 0.355i)27-s + (−0.360 + 0.932i)29-s + (0.877 − 0.479i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.999 + 0.0307i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.999 + 0.0307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.120672562 + 0.03265933684i\)
\(L(\frac12)\) \(\approx\) \(2.120672562 + 0.03265933684i\)
\(L(1)\) \(\approx\) \(1.233989864 - 0.1199519602i\)
\(L(1)\) \(\approx\) \(1.233989864 - 0.1199519602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.600 - 0.799i)T \)
5 \( 1 + (0.958 + 0.283i)T \)
7 \( 1 + (0.930 - 0.366i)T \)
11 \( 1 + (-0.889 + 0.457i)T \)
13 \( 1 + (0.418 + 0.908i)T \)
17 \( 1 + (0.998 - 0.0500i)T \)
19 \( 1 + (0.630 - 0.776i)T \)
23 \( 1 + (0.980 - 0.198i)T \)
29 \( 1 + (-0.360 + 0.932i)T \)
31 \( 1 + (0.877 - 0.479i)T \)
37 \( 1 + (0.999 - 0.0125i)T \)
41 \( 1 + (-0.106 + 0.994i)T \)
43 \( 1 + (-0.988 - 0.149i)T \)
47 \( 1 + (-0.131 + 0.991i)T \)
53 \( 1 + (0.630 + 0.776i)T \)
59 \( 1 + (-0.560 - 0.828i)T \)
61 \( 1 + (-0.900 + 0.435i)T \)
67 \( 1 + (0.349 - 0.937i)T \)
71 \( 1 + (-0.971 + 0.235i)T \)
73 \( 1 + (0.452 - 0.891i)T \)
79 \( 1 + (-0.831 + 0.554i)T \)
83 \( 1 + (-0.620 - 0.784i)T \)
89 \( 1 + (-0.993 - 0.112i)T \)
97 \( 1 + (0.999 + 0.0250i)T \)
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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

−18.24620852270111102299457257742, −17.81750973924703856804724326670, −16.99380749836861536259802629458, −16.626277169585254916800935604864, −15.71765435166815647010479511198, −15.16963970070509906515608527259, −14.47536352977984719262942902718, −13.71857440933205088586153035333, −13.0129273647046432897654529792, −12.20315603890494644302805066809, −11.52203271177077827251156899340, −10.79479027905314043529906078509, −10.13651856831116034764188645085, −9.75106865900389016183563421550, −8.662239001133966362506892737816, −8.25935053126678421579726763576, −7.30327080314291867427766434422, −6.075999052941413941307813573585, −5.52640383784183001722183949011, −5.30002564973672199971180214200, −4.42836247999306494154867448582, −3.32782794268780644885284628410, −2.708272947000090801241092080643, −1.50956819833835568974910617700, −0.76209903948550410323358950514, 1.05028346268072837459358567853, 1.51124665191538515485587713360, 2.414467645077725052955224673565, 3.11855566074982321793650208025, 4.69184296160234181148940990735, 4.9521642066616453420138085212, 5.825279511916053510429930943656, 6.51931290080688962025528213617, 7.307667784569594610481205660401, 7.73492587724596959943469332368, 8.68846478891638801185593957264, 9.55601080329612989122273672711, 10.34668233145865017864915821032, 11.00930518847636166170756425628, 11.49052297798728767327703049982, 12.32993715096194574283250558226, 13.21613469637576942332705466636, 13.51748510060031591723323427909, 14.30344551184383526529022883087, 14.8414390295371650550844968367, 15.89498784325516188955093367690, 16.85029912363381827109857422016, 17.05040557185843986565105089792, 17.93094509220189823883489433308, 18.46750682326187119019679091188

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