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Properties

Label 1-4024-4024.1189-r0-0-0
Degree $1$
Conductor $4024$
Sign $-0.433 + 0.901i$
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  + (−0.947 + 0.319i)3-s + (−0.864 + 0.501i)5-s + (0.337 − 0.941i)7-s + (0.795 − 0.605i)9-s + (0.996 − 0.0875i)11-s + (0.704 + 0.709i)13-s + (0.659 − 0.752i)15-s + (0.539 − 0.842i)17-s + (−0.474 − 0.880i)19-s + (−0.0187 + 0.999i)21-s + (−0.649 + 0.760i)23-s + (0.496 − 0.868i)25-s + (−0.560 + 0.828i)27-s + (−0.452 − 0.891i)29-s + (−0.831 + 0.554i)31-s + ⋯
L(s)  = 1  + (−0.947 + 0.319i)3-s + (−0.864 + 0.501i)5-s + (0.337 − 0.941i)7-s + (0.795 − 0.605i)9-s + (0.996 − 0.0875i)11-s + (0.704 + 0.709i)13-s + (0.659 − 0.752i)15-s + (0.539 − 0.842i)17-s + (−0.474 − 0.880i)19-s + (−0.0187 + 0.999i)21-s + (−0.649 + 0.760i)23-s + (0.496 − 0.868i)25-s + (−0.560 + 0.828i)27-s + (−0.452 − 0.891i)29-s + (−0.831 + 0.554i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-0.433 + 0.901i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ -0.433 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2916975088 + 0.4637506142i\)
\(L(\frac12)\) \(\approx\) \(0.2916975088 + 0.4637506142i\)
\(L(1)\) \(\approx\) \(0.6802417125 + 0.07674926972i\)
\(L(1)\) \(\approx\) \(0.6802417125 + 0.07674926972i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.947 + 0.319i)T \)
5 \( 1 + (-0.864 + 0.501i)T \)
7 \( 1 + (0.337 - 0.941i)T \)
11 \( 1 + (0.996 - 0.0875i)T \)
13 \( 1 + (0.704 + 0.709i)T \)
17 \( 1 + (0.539 - 0.842i)T \)
19 \( 1 + (-0.474 - 0.880i)T \)
23 \( 1 + (-0.649 + 0.760i)T \)
29 \( 1 + (-0.452 - 0.891i)T \)
31 \( 1 + (-0.831 + 0.554i)T \)
37 \( 1 + (-0.968 + 0.247i)T \)
41 \( 1 + (-0.528 + 0.848i)T \)
43 \( 1 + (0.990 - 0.137i)T \)
47 \( 1 + (-0.871 + 0.490i)T \)
53 \( 1 + (-0.474 + 0.880i)T \)
59 \( 1 + (-0.780 - 0.625i)T \)
61 \( 1 + (0.915 + 0.401i)T \)
67 \( 1 + (-0.659 - 0.752i)T \)
71 \( 1 + (0.0437 + 0.999i)T \)
73 \( 1 + (-0.999 + 0.0375i)T \)
79 \( 1 + (0.695 + 0.718i)T \)
83 \( 1 + (-0.677 + 0.735i)T \)
89 \( 1 + (-0.630 + 0.776i)T \)
97 \( 1 + (0.877 + 0.479i)T \)
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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

−18.25436699312317589406620384597, −17.57492203515441100197222952830, −16.83137348698577714814276809928, −16.30350597604878406052415406537, −15.69182043628397339280505277403, −14.87540248823017803119991206616, −14.3905960065858202757899082325, −13.10971772555798042168760762300, −12.52071844090068403316619943451, −12.17522709522000096819796111325, −11.51483772473029646621274974417, −10.82662139069326736935887934168, −10.17020258115450941069366399179, −9.02678985092339096765816858261, −8.44989879333293623217613647173, −7.83723444798128222550662715752, −7.00293128100830068519763703427, −6.02783214547610542371961088615, −5.699237873389891704072979413375, −4.82609612089261302320603125971, −3.985523265493841031758946384056, −3.405978938061584302487251373770, −1.85175335730728410435484076370, −1.45731058700920835596704101787, −0.22028436235288377580161001149, 0.91517596164031708557853838710, 1.66997031255994153831220478039, 3.16574108643445632742264315649, 3.92142114771747385488990189946, 4.28722699699234293896014886732, 5.123528713551732688369650094655, 6.16356360647692504310640441713, 6.805888560141635545576242790877, 7.27414977276297556387593085972, 8.10463914111513280842666190837, 9.14571631975578367640753597787, 9.76702077504234973312332591737, 10.69553900856539047432949671541, 11.20316945460846156623491313105, 11.62413616229473838394486925480, 12.22044817148972903530030812479, 13.261189564756424071344034698856, 14.03735361708427491468797762233, 14.58397200196511304569643040008, 15.49802305317435079283684179077, 16.02203419030720937256151025524, 16.66019368324677621796467402539, 17.26338244018410102445570493802, 17.914825793424552237384608727065, 18.65099275375482792659106536248

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