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Properties

Label 1-4024-4024.1107-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: $\chi_{4024} (1107, \cdot )$
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

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

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