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Properties

Label 1-39e2-1521.1109-r1-0-0
Degree $1$
Conductor $1521$
Sign $0.00413 + 0.999i$
Analytic cond. $163.454$
Root an. cond. $163.454$
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  + (−0.632 − 0.774i)2-s + (−0.200 + 0.979i)4-s + (0.799 + 0.600i)5-s + (−0.885 + 0.464i)7-s + (0.885 − 0.464i)8-s + (−0.0402 − 0.999i)10-s + (0.987 + 0.160i)11-s + (0.919 + 0.391i)14-s + (−0.919 − 0.391i)16-s + (0.0402 − 0.999i)17-s + (0.5 − 0.866i)19-s + (−0.748 + 0.663i)20-s + (−0.5 − 0.866i)22-s − 23-s + (0.278 + 0.960i)25-s + ⋯
L(s)  = 1  + (−0.632 − 0.774i)2-s + (−0.200 + 0.979i)4-s + (0.799 + 0.600i)5-s + (−0.885 + 0.464i)7-s + (0.885 − 0.464i)8-s + (−0.0402 − 0.999i)10-s + (0.987 + 0.160i)11-s + (0.919 + 0.391i)14-s + (−0.919 − 0.391i)16-s + (0.0402 − 0.999i)17-s + (0.5 − 0.866i)19-s + (−0.748 + 0.663i)20-s + (−0.5 − 0.866i)22-s − 23-s + (0.278 + 0.960i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00413 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00413 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.00413 + 0.999i$
Analytic conductor: \(163.454\)
Root analytic conductor: \(163.454\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1521,\ (1:\ ),\ 0.00413 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7607703590 + 0.7576341229i\)
\(L(\frac12)\) \(\approx\) \(0.7607703590 + 0.7576341229i\)
\(L(1)\) \(\approx\) \(0.8087623244 - 0.04242992586i\)
\(L(1)\) \(\approx\) \(0.8087623244 - 0.04242992586i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.632 - 0.774i)T \)
5 \( 1 + (0.799 + 0.600i)T \)
7 \( 1 + (-0.885 + 0.464i)T \)
11 \( 1 + (0.987 + 0.160i)T \)
17 \( 1 + (0.0402 - 0.999i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.632 + 0.774i)T \)
31 \( 1 + (-0.278 + 0.960i)T \)
37 \( 1 + (-0.692 - 0.721i)T \)
41 \( 1 + (0.568 + 0.822i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (0.948 + 0.316i)T \)
53 \( 1 + (-0.885 + 0.464i)T \)
59 \( 1 + (-0.919 + 0.391i)T \)
61 \( 1 + (0.885 + 0.464i)T \)
67 \( 1 + (0.748 - 0.663i)T \)
71 \( 1 + (-0.996 + 0.0804i)T \)
73 \( 1 + (0.354 + 0.935i)T \)
79 \( 1 + (0.948 + 0.316i)T \)
83 \( 1 + (-0.996 - 0.0804i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.120 - 0.992i)T \)
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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

−20.11845630713880425182002715463, −19.33216451756757267308040461706, −18.75164519735485755137997776040, −17.68488343405197411040631465374, −17.14758892192323585749105064392, −16.59607351679090486011566961114, −15.98851006409216307630898003263, −15.05895526104575811759539415487, −14.05315087394699242978490428233, −13.74182665024823043028934363382, −12.75136860186635406844115705056, −11.90763372945996030063629806832, −10.66677901496820101033868508266, −9.87693858698752233521721855445, −9.56391810315921963166307834773, −8.59309591193123457906059508922, −7.92419484332919351324183072894, −6.79873608078670413749871133994, −6.14630693055708326890169736317, −5.66983000228432546915608498158, −4.43486802043753934285217877779, −3.6401601405508597259202278602, −2.0580570254842002437442384943, −1.259388065280130894236667033973, −0.28377510613066313935023929795, 0.99559047133158561027373598514, 2.00756528864643462326027734680, 2.88151889447040497241567136245, 3.43448801517485530817276248577, 4.65234824505043342252375558902, 5.77295589235405242112637869205, 6.80622800784957321359498142606, 7.187556033105423198594358711322, 8.59519150607244641523602830035, 9.31149683150589609328328410790, 9.71724181981081761661037704466, 10.53848817825384264291758201675, 11.39954511487235418579764792786, 12.14773944315865760184960748901, 12.82015566356979115668898913737, 13.8292985145182880219687248906, 14.21575342957661248576460699806, 15.55573108216095507973753303228, 16.22928605863610002799508850519, 17.03955964369220403036265218264, 17.941359065370889345350615761606, 18.216122299500920232636614585776, 19.15125614450069045148200677683, 19.80303359004177415198805890758, 20.378528169278587460641266051148

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