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Properties

Label 1-4029-4029.1268-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.735 + 0.677i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (−0.975 − 0.219i)2-s + (0.903 + 0.428i)4-s + (0.287 − 0.957i)5-s + (−0.999 + 0.0100i)7-s + (−0.787 − 0.616i)8-s + (−0.491 + 0.871i)10-s + (0.229 − 0.973i)11-s + (0.0804 + 0.996i)13-s + (0.977 + 0.209i)14-s + (0.632 + 0.774i)16-s + (0.834 − 0.551i)19-s + (0.670 − 0.741i)20-s + (−0.437 + 0.899i)22-s + (0.991 − 0.130i)23-s + (−0.834 − 0.551i)25-s + (0.140 − 0.990i)26-s + ⋯
L(s)  = 1  + (−0.975 − 0.219i)2-s + (0.903 + 0.428i)4-s + (0.287 − 0.957i)5-s + (−0.999 + 0.0100i)7-s + (−0.787 − 0.616i)8-s + (−0.491 + 0.871i)10-s + (0.229 − 0.973i)11-s + (0.0804 + 0.996i)13-s + (0.977 + 0.209i)14-s + (0.632 + 0.774i)16-s + (0.834 − 0.551i)19-s + (0.670 − 0.741i)20-s + (−0.437 + 0.899i)22-s + (0.991 − 0.130i)23-s + (−0.834 − 0.551i)25-s + (0.140 − 0.990i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.735 + 0.677i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.735 + 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6838584267 + 0.2671675576i\)
\(L(\frac12)\) \(\approx\) \(0.6838584267 + 0.2671675576i\)
\(L(1)\) \(\approx\) \(0.6523762945 - 0.1025882316i\)
\(L(1)\) \(\approx\) \(0.6523762945 - 0.1025882316i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.975 - 0.219i)T \)
5 \( 1 + (0.287 - 0.957i)T \)
7 \( 1 + (-0.999 + 0.0100i)T \)
11 \( 1 + (0.229 - 0.973i)T \)
13 \( 1 + (0.0804 + 0.996i)T \)
19 \( 1 + (0.834 - 0.551i)T \)
23 \( 1 + (0.991 - 0.130i)T \)
29 \( 1 + (0.915 + 0.401i)T \)
31 \( 1 + (-0.907 + 0.419i)T \)
37 \( 1 + (0.655 + 0.755i)T \)
41 \( 1 + (-0.0302 + 0.999i)T \)
43 \( 1 + (-0.811 + 0.584i)T \)
47 \( 1 + (-0.316 + 0.948i)T \)
53 \( 1 + (0.990 + 0.140i)T \)
59 \( 1 + (-0.941 - 0.335i)T \)
61 \( 1 + (0.437 + 0.899i)T \)
67 \( 1 + (-0.885 + 0.464i)T \)
71 \( 1 + (0.871 - 0.491i)T \)
73 \( 1 + (-0.951 + 0.307i)T \)
83 \( 1 + (-0.941 + 0.335i)T \)
89 \( 1 + (-0.992 - 0.120i)T \)
97 \( 1 + (0.945 + 0.326i)T \)
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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

−18.35680213219059905025721253994, −17.81256249305984005133278365350, −17.12751905025316581655859858835, −16.484881050268547394251270139943, −15.567120060437082328589388523179, −15.23926990118048666466212349228, −14.550930962912331731466153220262, −13.704553234559512185287125712912, −12.83477970763322740887175854215, −12.11888576750094632838107242673, −11.33339886816171220298196691902, −10.47094873861949238541298227108, −10.0843859370461082830713167380, −9.52288633036709361700662297263, −8.79429390678321887795099581279, −7.72805482494941994526996314040, −7.20580420658092041769770375333, −6.68177364348302414538240756313, −5.856492524323027880148919305192, −5.29551058952155563067809864125, −3.80996571345383475989261463922, −3.07748308327194178847105683106, −2.434506344497251549750074320489, −1.52432760821217128005811920581, −0.33677919226369825057646352911, 0.96261442667057267421302429197, 1.39146202995618648829517905383, 2.693708197893257813260889013849, 3.19529761071950790494317283141, 4.19331155310431707121233049668, 5.1702175671007256120811249132, 6.126104442414432287387835915224, 6.62590900268841378045014446762, 7.41683477901140244794398385296, 8.43728085846144134772841604920, 8.91690369808975230231779635852, 9.44155938133689484295437722699, 10.01233051797835713515058058319, 10.99060828852052493040281369629, 11.61163339771545375508350419024, 12.25024051755753063236220395198, 13.07803796094795887668114040196, 13.50822503230881800149999250449, 14.48722640649159166512715416072, 15.52210487914865029337682643577, 16.27852598250856316959921178673, 16.46047456579282643771973833740, 17.030522353991748919742726682132, 17.945845650369738368688466457577, 18.569398927240316586729038938943

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