Properties

Label 1-4024-4024.389-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.976 + 0.213i$
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  + (−0.810 + 0.585i)3-s + (0.395 + 0.918i)5-s + (0.910 − 0.412i)7-s + (0.313 − 0.949i)9-s + (−0.192 + 0.981i)11-s + (−0.984 + 0.174i)13-s + (−0.858 − 0.512i)15-s + (0.889 − 0.457i)17-s + (0.831 − 0.554i)19-s + (−0.496 + 0.868i)21-s + (−0.325 − 0.945i)23-s + (−0.686 + 0.726i)25-s + (0.301 + 0.953i)27-s + (−0.407 − 0.913i)29-s + (0.0437 + 0.999i)31-s + ⋯
L(s)  = 1  + (−0.810 + 0.585i)3-s + (0.395 + 0.918i)5-s + (0.910 − 0.412i)7-s + (0.313 − 0.949i)9-s + (−0.192 + 0.981i)11-s + (−0.984 + 0.174i)13-s + (−0.858 − 0.512i)15-s + (0.889 − 0.457i)17-s + (0.831 − 0.554i)19-s + (−0.496 + 0.868i)21-s + (−0.325 − 0.945i)23-s + (−0.686 + 0.726i)25-s + (0.301 + 0.953i)27-s + (−0.407 − 0.913i)29-s + (0.0437 + 0.999i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.976 + 0.213i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.976 + 0.213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.449386419 + 0.1563998709i\)
\(L(\frac12)\) \(\approx\) \(1.449386419 + 0.1563998709i\)
\(L(1)\) \(\approx\) \(0.9522259623 + 0.2109620498i\)
\(L(1)\) \(\approx\) \(0.9522259623 + 0.2109620498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.810 + 0.585i)T \)
5 \( 1 + (0.395 + 0.918i)T \)
7 \( 1 + (0.910 - 0.412i)T \)
11 \( 1 + (-0.192 + 0.981i)T \)
13 \( 1 + (-0.984 + 0.174i)T \)
17 \( 1 + (0.889 - 0.457i)T \)
19 \( 1 + (0.831 - 0.554i)T \)
23 \( 1 + (-0.325 - 0.945i)T \)
29 \( 1 + (-0.407 - 0.913i)T \)
31 \( 1 + (0.0437 + 0.999i)T \)
37 \( 1 + (-0.118 - 0.992i)T \)
41 \( 1 + (0.974 + 0.223i)T \)
43 \( 1 + (-0.143 - 0.989i)T \)
47 \( 1 + (-0.894 - 0.446i)T \)
53 \( 1 + (0.831 + 0.554i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (0.418 - 0.908i)T \)
67 \( 1 + (0.858 - 0.512i)T \)
71 \( 1 + (-0.772 + 0.635i)T \)
73 \( 1 + (-0.507 + 0.861i)T \)
79 \( 1 + (0.640 - 0.768i)T \)
83 \( 1 + (0.649 - 0.760i)T \)
89 \( 1 + (0.877 - 0.479i)T \)
97 \( 1 + (-0.971 - 0.235i)T \)
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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

−18.133896131442104677982604926071, −17.910712397236981093789295353120, −16.99360245623332750136531963964, −16.63169885611452110843086708260, −16.01525308358300902804453498857, −15.012163996322916825754124238942, −14.22320849732460170018283118608, −13.59624926899388348923779061408, −12.865501670771088768703777185880, −12.22819384863602606158870952, −11.69047120590958487303411161370, −11.10685874956843637968311762851, −10.131006280548475032747867082833, −9.54271800098716699316881260300, −8.53916928546600518791935379469, −7.84890499670342322889200712909, −7.518757408672545657104936307383, −6.1815625152151822941838820950, −5.62921366384541132611968005354, −5.21995572034857670536351802595, −4.516170184747191244945549779767, −3.3402913394303926898172035637, −2.24113171770303734932075154602, −1.46212973711408471496800863520, −0.88007280312947248542450557086, 0.57191250919407491944816891482, 1.77895852441563173189603112786, 2.54521078907125146726262136570, 3.53077468480189414377814111830, 4.429674017510673649810211644584, 5.024455733485199603074560574354, 5.61704720187389972932933862258, 6.60425257404481952550371621980, 7.305250194445487421504348373047, 7.66358541946977528308669964360, 9.030370536608585387362288499242, 9.79374207559414765240369984529, 10.22634879443009160909773232198, 10.81611417370910714485215682644, 11.64476957197419846943884208348, 12.05136151250169006536116287161, 12.91646947592924705772261874648, 14.07406901558486357246284609484, 14.41358425083386175122812507908, 15.06082526939165119524420430920, 15.72764185435537246457633508272, 16.60162993547586278504437563085, 17.24976131095100468447872735561, 17.806176241416224020076998338135, 18.18103976281491364764016383584

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