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Properties

Label 1-2735-2735.1042-r0-0-0
Degree $1$
Conductor $2735$
Sign $-0.962 - 0.269i$
Analytic cond. $12.7012$
Root an. cond. $12.7012$
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  + (−0.171 + 0.985i)2-s + (−0.974 − 0.222i)3-s + (−0.940 − 0.338i)4-s + (0.386 − 0.922i)6-s + (0.877 + 0.479i)7-s + (0.495 − 0.868i)8-s + (0.900 + 0.433i)9-s + (0.885 − 0.464i)11-s + (0.842 + 0.539i)12-s + (−0.781 − 0.623i)13-s + (−0.623 + 0.781i)14-s + (0.770 + 0.636i)16-s + (−0.802 + 0.596i)17-s + (−0.582 + 0.813i)18-s + (0.418 − 0.908i)19-s + ⋯
L(s)  = 1  + (−0.171 + 0.985i)2-s + (−0.974 − 0.222i)3-s + (−0.940 − 0.338i)4-s + (0.386 − 0.922i)6-s + (0.877 + 0.479i)7-s + (0.495 − 0.868i)8-s + (0.900 + 0.433i)9-s + (0.885 − 0.464i)11-s + (0.842 + 0.539i)12-s + (−0.781 − 0.623i)13-s + (−0.623 + 0.781i)14-s + (0.770 + 0.636i)16-s + (−0.802 + 0.596i)17-s + (−0.582 + 0.813i)18-s + (0.418 − 0.908i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2735 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2735 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2735\)    =    \(5 \cdot 547\)
Sign: $-0.962 - 0.269i$
Analytic conductor: \(12.7012\)
Root analytic conductor: \(12.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2735} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2735,\ (0:\ ),\ -0.962 - 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03295945237 + 0.2397963658i\)
\(L(\frac12)\) \(\approx\) \(-0.03295945237 + 0.2397963658i\)
\(L(1)\) \(\approx\) \(0.5766123111 + 0.2367948377i\)
\(L(1)\) \(\approx\) \(0.5766123111 + 0.2367948377i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.171 + 0.985i)T \)
3 \( 1 + (-0.974 - 0.222i)T \)
7 \( 1 + (0.877 + 0.479i)T \)
11 \( 1 + (0.885 - 0.464i)T \)
13 \( 1 + (-0.781 - 0.623i)T \)
17 \( 1 + (-0.802 + 0.596i)T \)
19 \( 1 + (0.418 - 0.908i)T \)
23 \( 1 + (0.305 - 0.952i)T \)
29 \( 1 + (0.0862 + 0.996i)T \)
31 \( 1 + (-0.449 + 0.893i)T \)
37 \( 1 + (-0.688 - 0.725i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.103 - 0.994i)T \)
47 \( 1 + (-0.935 - 0.354i)T \)
53 \( 1 + (-0.957 + 0.289i)T \)
59 \( 1 + (-0.970 - 0.239i)T \)
61 \( 1 + (-0.962 + 0.272i)T \)
67 \( 1 + (-0.842 - 0.539i)T \)
71 \( 1 + (-0.978 - 0.205i)T \)
73 \( 1 + (0.957 + 0.289i)T \)
79 \( 1 + (-0.994 + 0.103i)T \)
83 \( 1 + (0.822 + 0.568i)T \)
89 \( 1 + (0.962 + 0.272i)T \)
97 \( 1 + (0.402 + 0.915i)T \)
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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

−18.84562603989659368403521380611, −18.11776979338851030340408720331, −17.43248849490339032652026602448, −17.0908223776536702683553331662, −16.41988820326406910137378023223, −15.25974281883213587860440098244, −14.536630214851509348348850802593, −13.77044775269485338017488688684, −13.05069286189363424918357243001, −11.98791620341000000450353706593, −11.71451769799672794133903123532, −11.2153018158855223812758242147, −10.30336461846079774604841829369, −9.64769856402604747200228330415, −9.16344094815014071609167322809, −7.922137942692180407860515669692, −7.300846493717604265253748865429, −6.371239548638066153697803179583, −5.278012245450307595010265692937, −4.56891192782013975646509329365, −4.202666928013979027487354343471, −3.177010492737010263352428840, −1.72083654721492435254454983103, −1.51407784820514717668285765267, −0.104457482373102390527893342572, 1.07731695212749622908057404307, 1.93553965166581291710225827549, 3.40668420423387870171722113631, 4.624336854596518167239796146280, 4.96692555723647123756373279262, 5.70351358729430498599612707773, 6.58328647726985324238278003300, 7.038910561205823481507753937847, 7.92701294683652067833482507752, 8.75920641641387537674111230994, 9.27300475612142247523209669254, 10.56113367630664790976041572364, 10.82056763194704565337352676722, 11.94244900891697464292145906823, 12.467319544022536207660204163717, 13.31432153654157152093240786564, 14.1450294471897609371033239038, 14.85043815786241534012440217492, 15.45730394549023771919136131604, 16.210070695942233943462917233380, 16.99236946220217058470483889383, 17.447807738743478016376973831450, 17.99413548497999817612929118066, 18.599897599253325655975495756563, 19.44699767573012752290286985307

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