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Properties

Label 1-2668-2668.1087-r0-0-0
Degree $1$
Conductor $2668$
Sign $-0.486 - 0.873i$
Analytic cond. $12.3901$
Root an. cond. $12.3901$
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  + (0.852 − 0.523i)3-s + (0.979 + 0.202i)5-s + (−0.742 − 0.670i)7-s + (0.452 − 0.891i)9-s + (−0.983 + 0.182i)11-s + (0.377 − 0.925i)13-s + (0.940 − 0.339i)15-s + (−0.989 + 0.142i)17-s + (0.953 − 0.301i)19-s + (−0.983 − 0.182i)21-s + (0.917 + 0.396i)25-s + (−0.0815 − 0.996i)27-s + (0.699 − 0.714i)31-s + (−0.742 + 0.670i)33-s + (−0.591 − 0.806i)35-s + ⋯
L(s)  = 1  + (0.852 − 0.523i)3-s + (0.979 + 0.202i)5-s + (−0.742 − 0.670i)7-s + (0.452 − 0.891i)9-s + (−0.983 + 0.182i)11-s + (0.377 − 0.925i)13-s + (0.940 − 0.339i)15-s + (−0.989 + 0.142i)17-s + (0.953 − 0.301i)19-s + (−0.983 − 0.182i)21-s + (0.917 + 0.396i)25-s + (−0.0815 − 0.996i)27-s + (0.699 − 0.714i)31-s + (−0.742 + 0.670i)33-s + (−0.591 − 0.806i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2668\)    =    \(2^{2} \cdot 23 \cdot 29\)
Sign: $-0.486 - 0.873i$
Analytic conductor: \(12.3901\)
Root analytic conductor: \(12.3901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2668} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2668,\ (0:\ ),\ -0.486 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066892082 - 1.814473424i\)
\(L(\frac12)\) \(\approx\) \(1.066892082 - 1.814473424i\)
\(L(1)\) \(\approx\) \(1.308389260 - 0.5563865750i\)
\(L(1)\) \(\approx\) \(1.308389260 - 0.5563865750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.852 - 0.523i)T \)
5 \( 1 + (0.979 + 0.202i)T \)
7 \( 1 + (-0.742 - 0.670i)T \)
11 \( 1 + (-0.983 + 0.182i)T \)
13 \( 1 + (0.377 - 0.925i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (0.953 - 0.301i)T \)
31 \( 1 + (0.699 - 0.714i)T \)
37 \( 1 + (-0.891 - 0.452i)T \)
41 \( 1 + (0.909 + 0.415i)T \)
43 \( 1 + (-0.699 - 0.714i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (-0.0611 + 0.998i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (0.122 + 0.992i)T \)
67 \( 1 + (0.182 - 0.983i)T \)
71 \( 1 + (-0.999 + 0.0407i)T \)
73 \( 1 + (0.359 - 0.933i)T \)
79 \( 1 + (0.574 - 0.818i)T \)
83 \( 1 + (0.768 + 0.639i)T \)
89 \( 1 + (-0.940 - 0.339i)T \)
97 \( 1 + (-0.999 + 0.0203i)T \)
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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

−19.51758353276847244166114188052, −18.894476827093928421011403829362, −18.22378096325859687210591454504, −17.54773682552007265454997797905, −16.29686865829399160782694763748, −16.115451193480088897059219788004, −15.43039987750426289503851866247, −14.46523568349856182045617955120, −13.829811112096821110361794482283, −13.27079072222457095453392789479, −12.714375647243554607289972070064, −11.62744462698741985095860028562, −10.72283598327127379399091735338, −9.89976392453329504148037019412, −9.51003302829770804462310330401, −8.73625134626700300037642486678, −8.24224181306326686014592144781, −7.02305116174265993499178270068, −6.34931794960126935023962773619, −5.36189169090480820383464870190, −4.840601794238989315883061451058, −3.73534032731333340827245248581, −2.84831313763368891143040578206, −2.33624511264729098417230017701, −1.42239339112007031969566004825, 0.53874103772184680366677311896, 1.593331077796168354652765230462, 2.572389788766834659055699148194, 3.02795135819755032624225128476, 3.943708960545733743196728063204, 5.078126228337525602730951624419, 5.96258651239458729422579043134, 6.69861900583736701717346171266, 7.35828223153802196643625335555, 8.10289796282632700794391050141, 8.98577251582295582836856712391, 9.69361761206844378231998501227, 10.288633269118965595929818163341, 10.941387259802163549409777983528, 12.20876244144261059592266949546, 13.02436365826820464411259406718, 13.46896590306437675226094009149, 13.72586740552638543885372372422, 14.80760342547649614640030404589, 15.47531177032480118930187711104, 16.11179337907516758422816728080, 17.16480468731356423487443669820, 17.98177694107623563742610483013, 18.136208035850130277498152668355, 19.15947959623506121476768590426

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