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Properties

Label 1-2667-2667.524-r0-0-0
Degree $1$
Conductor $2667$
Sign $0.945 + 0.324i$
Analytic cond. $12.3854$
Root an. cond. $12.3854$
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  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.623 − 0.781i)5-s + (0.222 + 0.974i)8-s + (0.900 − 0.433i)10-s + (0.900 − 0.433i)11-s + (−0.623 + 0.781i)13-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 19-s + 20-s + 22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.900 + 0.433i)26-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.623 − 0.781i)5-s + (0.222 + 0.974i)8-s + (0.900 − 0.433i)10-s + (0.900 − 0.433i)11-s + (−0.623 + 0.781i)13-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 19-s + 20-s + 22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.900 + 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.945 + 0.324i$
Analytic conductor: \(12.3854\)
Root analytic conductor: \(12.3854\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (0:\ ),\ 0.945 + 0.324i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.660499891 + 0.6109350028i\)
\(L(\frac12)\) \(\approx\) \(3.660499891 + 0.6109350028i\)
\(L(1)\) \(\approx\) \(2.085023621 + 0.3458238402i\)
\(L(1)\) \(\approx\) \(2.085023621 + 0.3458238402i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 - T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.623 - 0.781i)T \)
37 \( 1 + T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (0.222 + 0.974i)T \)
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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

−19.5877456933221023406299342433, −18.6637001412347483221098073438, −17.98825372725391243372341079894, −17.0643555222859922232230720938, −16.55869933409077247529153162310, −15.2369259974298219217963826735, −14.828181246490301779316122392378, −14.469257176649801242160698814, −13.59774979116816109130068110850, −12.670064589116345992420828893613, −12.45299255813205221079721816650, −11.34231550649881633159942929974, −10.673940177470542871482810318801, −10.15904862181421332752946106186, −9.43820216109497337830933530041, −8.41642175448888900647533459621, −7.08481803494016224481724450472, −6.83825055177278413556938862256, −5.857582347913620732762041517420, −5.28878529610049431045035374271, −4.290820053500175931811821132027, −3.50568484967077104232071459533, −2.72507691502798034222597374372, −1.9805300952989067049247574263, −1.08407848560258759636046062142, 0.97361725105948060713978512798, 2.0657423182820279283554769062, 2.76378438214110181845585286084, 4.005643168598775404658882071679, 4.43578588033949679215619199715, 5.36402664173432367544355811856, 5.97891025090630527940127613177, 6.72214143618823425920862122517, 7.51264606179925241122120306692, 8.40931372482143631445718395205, 9.203346814884627677017044685196, 9.750189766883019786870074600241, 11.05078342174227401621170709809, 11.65106961186889502906610337384, 12.35599928850264257423066533133, 13.03716463837541345832855754195, 13.69069595388490050686965144814, 14.37561266177203034169060284099, 14.865896206469084219656781012163, 15.90753622721397845146035713085, 16.56150115654972257655414515502, 17.058614143737340318414630045557, 17.49622270779023628392360330113, 18.73537711474249888460245935125, 19.48204210053096815624773837973

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