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Properties

Label 1-1976-1976.1283-r0-0-0
Degree $1$
Conductor $1976$
Sign $-0.242 - 0.970i$
Analytic cond. $9.17650$
Root an. cond. $9.17650$
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  + (0.939 − 0.342i)3-s + (−0.766 − 0.642i)5-s − 7-s + (0.766 − 0.642i)9-s + 11-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.939 + 0.342i)21-s + (0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.939 − 0.342i)33-s + (0.766 + 0.642i)35-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (−0.766 − 0.642i)5-s − 7-s + (0.766 − 0.642i)9-s + 11-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.939 + 0.342i)21-s + (0.939 + 0.342i)23-s + (0.173 + 0.984i)25-s + (0.5 − 0.866i)27-s + (0.173 + 0.984i)29-s + (−0.5 − 0.866i)31-s + (0.939 − 0.342i)33-s + (0.766 + 0.642i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1976 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1976\)    =    \(2^{3} \cdot 13 \cdot 19\)
Sign: $-0.242 - 0.970i$
Analytic conductor: \(9.17650\)
Root analytic conductor: \(9.17650\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1976} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1976,\ (0:\ ),\ -0.242 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.071550431 - 1.372508717i\)
\(L(\frac12)\) \(\approx\) \(1.071550431 - 1.372508717i\)
\(L(1)\) \(\approx\) \(1.150452437 - 0.4434284949i\)
\(L(1)\) \(\approx\) \(1.150452437 - 0.4434284949i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−19.93500138393415837896576304122, −19.44325377158038079795732504284, −19.07159287423596043843365484207, −18.30711601650367530170342335734, −17.04334984220487518577637395605, −16.491158785727942053779645305263, −15.53062609053705048655623729186, −15.171062453751473822207988006964, −14.42158454430434680908419484850, −13.73618939199810874475628501371, −12.832873200553201997311372449852, −12.17902813152766991821016089026, −11.20399165396149588843521130423, −10.36083808729021006236856967754, −9.80652904818342725372482083771, −8.83149712890283064283952787363, −8.36929788051496690455769604274, −7.23380564286606012483990077674, −6.8216034225858839280432202121, −5.85801565163524532964380625024, −4.46241247295878256436890155312, −3.835967960749246166377055540217, −3.21815058631509205075947629362, −2.47203893808739852517691057214, −1.20519765253358528045176400233, 0.58635811710371926827487799414, 1.51558004646008771194086472186, 2.74806858826753374644602346288, 3.50005362517626070715189901536, 4.07942566508745739962974981885, 5.11271666931337089543398440235, 6.26377453722004975960507385433, 7.19134283585408056132245344790, 7.51609204840871707095464406027, 8.77635870409176648114529875687, 9.09565699064247500936182974792, 9.71734314796761441694816522419, 10.914396441831932618770941889226, 11.91229530375275580652309702130, 12.43229722991977051545314536101, 13.14444479952988962513415386543, 13.80979272407644549337566027817, 14.70009289677127242703316000539, 15.323831819238352974972980722297, 16.17566811320433954924665084261, 16.5921903027108271399816345984, 17.631218188565369851362926137779, 18.63709746235946413094714534697, 19.19628526969917769823205800568, 19.7723768367468647860431155571

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