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Properties

Label 1-1520-1520.1219-r0-0-0
Degree $1$
Conductor $1520$
Sign $-0.608 - 0.793i$
Analytic cond. $7.05885$
Root an. cond. $7.05885$
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  + (−0.642 − 0.766i)3-s + (0.5 − 0.866i)7-s + (−0.173 + 0.984i)9-s + (0.866 − 0.5i)11-s + (−0.642 + 0.766i)13-s + (−0.173 − 0.984i)17-s + (−0.984 + 0.173i)21-s + (0.939 − 0.342i)23-s + (0.866 − 0.5i)27-s + (−0.984 − 0.173i)29-s + (−0.5 + 0.866i)31-s + (−0.939 − 0.342i)33-s − i·37-s + 39-s + (0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)3-s + (0.5 − 0.866i)7-s + (−0.173 + 0.984i)9-s + (0.866 − 0.5i)11-s + (−0.642 + 0.766i)13-s + (−0.173 − 0.984i)17-s + (−0.984 + 0.173i)21-s + (0.939 − 0.342i)23-s + (0.866 − 0.5i)27-s + (−0.984 − 0.173i)29-s + (−0.5 + 0.866i)31-s + (−0.939 − 0.342i)33-s − i·37-s + 39-s + (0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(7.05885\)
Root analytic conductor: \(7.05885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1520,\ (0:\ ),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4932367457 - 0.9993628553i\)
\(L(\frac12)\) \(\approx\) \(0.4932367457 - 0.9993628553i\)
\(L(1)\) \(\approx\) \(0.8144624232 - 0.3897838697i\)
\(L(1)\) \(\approx\) \(0.8144624232 - 0.3897838697i\)

Euler product

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

−20.99350478051119190652137250785, −20.233140024879926153517891906091, −19.42479019466058245867476358250, −18.487685856502985610521181496290, −17.649620714591327677973426276245, −17.170973347407330528974004191646, −16.50925778180110411961572477569, −15.265066882624968886472590876846, −15.10579955664929099865798594358, −14.47331001727470214310660262631, −13.03394499410899144511409604559, −12.40920490454293205168702193178, −11.60576056009308433369220123469, −11.05324118768598034706131206808, −10.10489032399700660591612334053, −9.38137799842429446632833241195, −8.72044950299954078513377558864, −7.68996847196477916568035387089, −6.658132996504412197306878402107, −5.77614250933124068541169711374, −5.172358084298032401213602343916, −4.318898820695111514595483959977, −3.45930856337848328996481187441, −2.327888359328983626091101934, −1.200594159274192172050846465247, 0.50719657293311034055652175809, 1.44483992777328841422197203080, 2.36232234736144817493853912906, 3.66019119059126914199848941675, 4.63557088967557371801158041277, 5.32563077442436982080952941122, 6.42483037689576502294569840647, 7.11413437107265899353369812531, 7.5719124856473357058723523824, 8.74405761263748612364932414087, 9.49238726673612693240691316214, 10.70399038740891042284242272214, 11.213499472734777791217345159007, 11.87473506222320007119882243921, 12.68314168217954402338519967427, 13.607489788321370342168326139387, 14.131301576542124576995718413532, 14.82005461172103396254223982170, 16.24183816477816901999192035915, 16.67494088781602007742867044641, 17.305879857048232473638059224106, 18.02814035439195201148282139804, 18.82576306389111005003858351575, 19.528577269528604696162224915024, 20.14512573898411497585883424430

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