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Properties

Label 1-1520-1520.187-r0-0-0
Degree $1$
Conductor $1520$
Sign $-0.501 + 0.865i$
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  + (0.766 − 0.642i)3-s + (−0.866 − 0.5i)7-s + (0.173 − 0.984i)9-s + (−0.866 + 0.5i)11-s + (−0.766 − 0.642i)13-s + (−0.984 + 0.173i)17-s + (−0.984 + 0.173i)21-s + (0.342 + 0.939i)23-s + (−0.5 − 0.866i)27-s + (0.984 + 0.173i)29-s + (0.5 − 0.866i)31-s + (−0.342 + 0.939i)33-s − 37-s − 39-s + (−0.766 + 0.642i)41-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (−0.866 − 0.5i)7-s + (0.173 − 0.984i)9-s + (−0.866 + 0.5i)11-s + (−0.766 − 0.642i)13-s + (−0.984 + 0.173i)17-s + (−0.984 + 0.173i)21-s + (0.342 + 0.939i)23-s + (−0.5 − 0.866i)27-s + (0.984 + 0.173i)29-s + (0.5 − 0.866i)31-s + (−0.342 + 0.939i)33-s − 37-s − 39-s + (−0.766 + 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.501 + 0.865i$
Analytic conductor: \(7.05885\)
Root analytic conductor: \(7.05885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1520,\ (0:\ ),\ -0.501 + 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05147437295 + 0.08931036533i\)
\(L(\frac12)\) \(\approx\) \(0.05147437295 + 0.08931036533i\)
\(L(1)\) \(\approx\) \(0.8567806155 - 0.2291021293i\)
\(L(1)\) \(\approx\) \(0.8567806155 - 0.2291021293i\)

Euler product

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

−20.40943924029537012231519576050, −19.40849104141891036501213809447, −19.18022598194981237056461136275, −18.28769499818979884938082654887, −17.235291391203167331387470749494, −16.32280296707518537178440619439, −15.77785336206176844495904824110, −15.27644252616694821853691747345, −14.22969960123704997435228515773, −13.699851176392021410990708822931, −12.82098564787779222128600989955, −12.10708196423879552912278004903, −10.93099544265038632581399245653, −10.30700526047317175911678006575, −9.50324346461589475662414501236, −8.80835319401696632088557427160, −8.204533195814259710221098405658, −7.07187800732884245758545640458, −6.340674886357872922050720461195, −5.10255046695127519152889138881, −4.57995872676852249931291934071, −3.375239349250734709218120324421, −2.7459017488402887162558341413, −2.00235237588493098385270331039, −0.03224677630885446992416083747, 1.31030589556989458896552541822, 2.50982668811286874630534230058, 3.00722201070733303710757586267, 4.05433168904335820358356557177, 5.04006663502062291823327836590, 6.20759954519168637833328862436, 6.9676277624028003049841914990, 7.612828145049978035386972732478, 8.34508744143555953803224343245, 9.3865900359969571946130716307, 9.94942063369193266720879181734, 10.77397450916511552244354240697, 11.983909378256267315298679763354, 12.704595637881869312585537996914, 13.29693108037233935858687180499, 13.8002080551646902405009854101, 14.94514542770871077673281396145, 15.416141407063310744021503137123, 16.20411822227998208261730029658, 17.51855272121205160056659371792, 17.65737266548583412661403965950, 18.81255667292193845015226880236, 19.42025555712123750667147942734, 20.00574096406972572314198313261, 20.59791114908940897068190507423

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