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Properties

Label 1-4024-4024.1763-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.573 + 0.819i$
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  + (−0.971 − 0.235i)3-s + (0.810 + 0.585i)5-s + (0.915 + 0.401i)7-s + (0.889 + 0.457i)9-s + (0.407 + 0.913i)11-s + (0.590 − 0.806i)13-s + (−0.649 − 0.760i)15-s + (−0.360 + 0.932i)17-s + (0.787 + 0.615i)19-s + (−0.795 − 0.605i)21-s + (−0.0937 − 0.995i)23-s + (0.313 + 0.949i)25-s + (−0.756 − 0.654i)27-s + (−0.845 + 0.533i)29-s + (0.852 + 0.523i)31-s + ⋯
L(s)  = 1  + (−0.971 − 0.235i)3-s + (0.810 + 0.585i)5-s + (0.915 + 0.401i)7-s + (0.889 + 0.457i)9-s + (0.407 + 0.913i)11-s + (0.590 − 0.806i)13-s + (−0.649 − 0.760i)15-s + (−0.360 + 0.932i)17-s + (0.787 + 0.615i)19-s + (−0.795 − 0.605i)21-s + (−0.0937 − 0.995i)23-s + (0.313 + 0.949i)25-s + (−0.756 − 0.654i)27-s + (−0.845 + 0.533i)29-s + (0.852 + 0.523i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.573 + 0.819i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.573 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.764890435 + 0.9187126791i\)
\(L(\frac12)\) \(\approx\) \(1.764890435 + 0.9187126791i\)
\(L(1)\) \(\approx\) \(1.142012752 + 0.2238543906i\)
\(L(1)\) \(\approx\) \(1.142012752 + 0.2238543906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.971 - 0.235i)T \)
5 \( 1 + (0.810 + 0.585i)T \)
7 \( 1 + (0.915 + 0.401i)T \)
11 \( 1 + (0.407 + 0.913i)T \)
13 \( 1 + (0.590 - 0.806i)T \)
17 \( 1 + (-0.360 + 0.932i)T \)
19 \( 1 + (0.787 + 0.615i)T \)
23 \( 1 + (-0.0937 - 0.995i)T \)
29 \( 1 + (-0.845 + 0.533i)T \)
31 \( 1 + (0.852 + 0.523i)T \)
37 \( 1 + (0.955 - 0.295i)T \)
41 \( 1 + (0.831 - 0.554i)T \)
43 \( 1 + (-0.894 - 0.446i)T \)
47 \( 1 + (0.999 + 0.0125i)T \)
53 \( 1 + (-0.787 + 0.615i)T \)
59 \( 1 + (-0.131 - 0.991i)T \)
61 \( 1 + (0.180 - 0.983i)T \)
67 \( 1 + (-0.649 + 0.760i)T \)
71 \( 1 + (0.838 + 0.544i)T \)
73 \( 1 + (0.265 - 0.964i)T \)
79 \( 1 + (-0.0187 + 0.999i)T \)
83 \( 1 + (-0.934 + 0.355i)T \)
89 \( 1 + (0.905 - 0.424i)T \)
97 \( 1 + (0.824 + 0.565i)T \)
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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

−18.13744308838783342185948556918, −17.66586545247178513689135632750, −16.9537865814096206635027628846, −16.50801030087840000333635349996, −15.89994918459631026464460242257, −15.059931148048046504812733411341, −14.02130025838101506996824966094, −13.603242479251877732078408786476, −13.08418273820239803152914053313, −11.820042606381039345507189234511, −11.48488607383933654504468653230, −11.070241586728352377254667956296, −10.02163484077696236804396301379, −9.397698757175596267115539530508, −8.861763826095237449217969607544, −7.824421648081007093360374994, −7.06012618215446334005106597316, −6.12204990051685113493897808426, −5.75165924913107110363917641019, −4.78614701633172280069290997227, −4.45264400948837604589288632635, −3.44818614009369731219135963679, −2.20276159249613553804858835991, −1.24587409070585526869861195443, −0.79653726297077376664163360337, 1.05983258746273153936089837303, 1.75057051348245295619057014662, 2.39325177078307368302067593967, 3.61222771660992731363679439930, 4.53909564787414166969923970901, 5.26369208390725034015524272953, 5.92773547892060543043957710306, 6.43595296364358443557881325590, 7.30796276119986438355333438175, 7.96256149941351464900416082188, 8.88686032154161460477230702806, 9.80122204660295924711911220813, 10.46311246404161141932433203693, 10.93921497017002987990090724425, 11.63341248778241613089301975356, 12.5472271990534196050757545023, 12.818670831862476999271277881925, 13.89700092109448545270653690028, 14.46458341910875503546538018311, 15.20784004878376879698034825305, 15.78396031842988087675995696347, 16.87286755817750014592923557154, 17.32032472692479367317634279476, 17.90446986011046891964186528506, 18.36727956778086556091223835257

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