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Properties

Label 1-1340-1340.1003-r0-0-0
Degree $1$
Conductor $1340$
Sign $0.731 - 0.681i$
Analytic cond. $6.22293$
Root an. cond. $6.22293$
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  + (0.540 − 0.841i)3-s + (0.814 + 0.580i)7-s + (−0.415 − 0.909i)9-s + (0.786 − 0.618i)11-s + (0.971 + 0.235i)13-s + (−0.189 − 0.981i)17-s + (0.580 + 0.814i)19-s + (0.928 − 0.371i)21-s + (0.458 + 0.888i)23-s + (−0.989 − 0.142i)27-s + (0.5 + 0.866i)29-s + (−0.235 − 0.971i)31-s + (−0.0950 − 0.995i)33-s + (0.866 + 0.5i)37-s + (0.723 − 0.690i)39-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (0.814 + 0.580i)7-s + (−0.415 − 0.909i)9-s + (0.786 − 0.618i)11-s + (0.971 + 0.235i)13-s + (−0.189 − 0.981i)17-s + (0.580 + 0.814i)19-s + (0.928 − 0.371i)21-s + (0.458 + 0.888i)23-s + (−0.989 − 0.142i)27-s + (0.5 + 0.866i)29-s + (−0.235 − 0.971i)31-s + (−0.0950 − 0.995i)33-s + (0.866 + 0.5i)37-s + (0.723 − 0.690i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $0.731 - 0.681i$
Analytic conductor: \(6.22293\)
Root analytic conductor: \(6.22293\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1340,\ (0:\ ),\ 0.731 - 0.681i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.199009359 - 0.8656605371i\)
\(L(\frac12)\) \(\approx\) \(2.199009359 - 0.8656605371i\)
\(L(1)\) \(\approx\) \(1.477550427 - 0.3741521352i\)
\(L(1)\) \(\approx\) \(1.477550427 - 0.3741521352i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good3 \( 1 + (0.540 - 0.841i)T \)
7 \( 1 + (0.814 + 0.580i)T \)
11 \( 1 + (0.786 - 0.618i)T \)
13 \( 1 + (0.971 + 0.235i)T \)
17 \( 1 + (-0.189 - 0.981i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
23 \( 1 + (0.458 + 0.888i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.235 - 0.971i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.327 + 0.945i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + (0.998 + 0.0475i)T \)
53 \( 1 + (-0.755 - 0.654i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (-0.981 - 0.189i)T \)
73 \( 1 + (0.618 - 0.786i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (0.371 - 0.928i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−20.95323026191928433177944205987, −20.18807538054267100733750918354, −19.85254265190137729658820755967, −18.81581638683479434279682295025, −17.78628602918555842820163244296, −17.17798234061412661071119637438, −16.44881120201404581953929265168, −15.429457368997896015369162756634, −15.02616278040566404368260002050, −14.08194234934782980453631499588, −13.65267156532311529744362729966, −12.558871879776150478254811796421, −11.50353288691555850968005884155, −10.75366498554325804607923483171, −10.2632706839906401489482082093, −9.12180459462451530747976098680, −8.62137541175390494914842468833, −7.75553103536208577721941874625, −6.84124326559303830030704029297, −5.75044309218308518457908607692, −4.664849004796824876903674486381, −4.17840947259759214691695306306, −3.30781610973758820236187471150, −2.15879723308333180051035423878, −1.14689483949335944844976051679, 1.12169654874776603104947811251, 1.67462348588413058615020070552, 2.89957941409684161873930008930, 3.59600180999329585474962161894, 4.81908080389548208849841728707, 5.88444708400768669629498810924, 6.49626402730932808467367714773, 7.58009005386746666237531002182, 8.188702894499944087263574016654, 9.012226248355049269588451125198, 9.54408798373558925413055383239, 11.122706117610153319064940580939, 11.55630233208352547001159543300, 12.249523950321946278597308090553, 13.33491943019427257920468351643, 13.88554996783056827215825537160, 14.54640847836548950662079943627, 15.30938530279381232043012421993, 16.27769346404976159071777658845, 17.11647395062335591964249274089, 18.22093073634003778963926515278, 18.34670399846998341015375318403, 19.19499073403550499732394692357, 20.11210843727323660535070665618, 20.670044257310921825011044758671

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