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Properties

Label 1-4024-4024.3405-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.949 - 0.314i$
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  + (0.977 − 0.211i)3-s + (−0.0187 + 0.999i)5-s + (0.0437 − 0.999i)7-s + (0.910 − 0.412i)9-s + (0.704 + 0.709i)11-s + (0.686 − 0.726i)13-s + (0.192 + 0.981i)15-s + (−0.915 + 0.401i)17-s + (0.965 − 0.259i)19-s + (−0.168 − 0.985i)21-s + (−0.0812 − 0.996i)23-s + (−0.999 − 0.0375i)25-s + (0.803 − 0.595i)27-s + (0.883 − 0.468i)29-s + (−0.549 + 0.835i)31-s + ⋯
L(s)  = 1  + (0.977 − 0.211i)3-s + (−0.0187 + 0.999i)5-s + (0.0437 − 0.999i)7-s + (0.910 − 0.412i)9-s + (0.704 + 0.709i)11-s + (0.686 − 0.726i)13-s + (0.192 + 0.981i)15-s + (−0.915 + 0.401i)17-s + (0.965 − 0.259i)19-s + (−0.168 − 0.985i)21-s + (−0.0812 − 0.996i)23-s + (−0.999 − 0.0375i)25-s + (0.803 − 0.595i)27-s + (0.883 − 0.468i)29-s + (−0.549 + 0.835i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (3405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.949 - 0.314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.001580693 - 0.4838157464i\)
\(L(\frac12)\) \(\approx\) \(3.001580693 - 0.4838157464i\)
\(L(1)\) \(\approx\) \(1.673129416 - 0.09031770275i\)
\(L(1)\) \(\approx\) \(1.673129416 - 0.09031770275i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.977 - 0.211i)T \)
5 \( 1 + (-0.0187 + 0.999i)T \)
7 \( 1 + (0.0437 - 0.999i)T \)
11 \( 1 + (0.704 + 0.709i)T \)
13 \( 1 + (0.686 - 0.726i)T \)
17 \( 1 + (-0.915 + 0.401i)T \)
19 \( 1 + (0.965 - 0.259i)T \)
23 \( 1 + (-0.0812 - 0.996i)T \)
29 \( 1 + (0.883 - 0.468i)T \)
31 \( 1 + (-0.549 + 0.835i)T \)
37 \( 1 + (0.630 - 0.776i)T \)
41 \( 1 + (0.955 - 0.295i)T \)
43 \( 1 + (0.325 + 0.945i)T \)
47 \( 1 + (0.0937 + 0.995i)T \)
53 \( 1 + (0.965 + 0.259i)T \)
59 \( 1 + (-0.980 - 0.198i)T \)
61 \( 1 + (-0.838 + 0.544i)T \)
67 \( 1 + (-0.192 + 0.981i)T \)
71 \( 1 + (0.384 - 0.923i)T \)
73 \( 1 + (-0.943 - 0.331i)T \)
79 \( 1 + (0.600 - 0.799i)T \)
83 \( 1 + (-0.407 - 0.913i)T \)
89 \( 1 + (0.143 - 0.989i)T \)
97 \( 1 + (-0.205 + 0.978i)T \)
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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

−18.45341447892479466769197071071, −18.11896409375546395260175616505, −16.921582225092776635013017941998, −16.35512723415575980684899115203, −15.68290520018375705280093982878, −15.33050780484925293331405394865, −14.26908939263367660882712233441, −13.73412369288999499028577301657, −13.25140160165281454532474179213, −12.34598507887378276280145706752, −11.67806589940595145126210137387, −11.127225749693702761006816835965, −9.84446004465352392695459085282, −9.21317168312758606515955905951, −8.91586832589858222867711454431, −8.304608499011675529670935596847, −7.52808567484626944313430704330, −6.52432584544672521739278699977, −5.71681143477377268433006267807, −4.96636684931921426027078170997, −4.128096656743890452608413629218, −3.52528091061468832154230044059, −2.58293910669247507522998826951, −1.74945676523870756079114615336, −1.049496285434852859773848097947, 0.861211225724474733547352534571, 1.71609349567626159216879106365, 2.669844735180692209538614216452, 3.25426599790771912917064989237, 4.11788377961882228682041529635, 4.53006067288512064314641299396, 6.10087601612174940635357641097, 6.57862605327713514446518680983, 7.46721811614168082735447195222, 7.65925676038203414065644425734, 8.75156437147618338612284877495, 9.37402515467833946762256815969, 10.28885752923548888757181007041, 10.628362902703552686286321385514, 11.46999980309975573224096158217, 12.4544909834604405342808653679, 13.12600707927682179008086382312, 13.807970733231468274701158362196, 14.369582074436062814871169206200, 14.815488316324415215903099670313, 15.65793106656399381777414801353, 16.15533529409782970036775432727, 17.348363202633471303973703699557, 17.984351981475944058073635577185, 18.21523932827118753139168172344

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