Properties

Label 1-2280-2280.1613-r0-0-0
Degree $1$
Conductor $2280$
Sign $-0.0457 - 0.998i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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  + (0.866 + 0.5i)7-s + (−0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.342 − 0.939i)17-s + (0.642 − 0.766i)23-s + (0.939 − 0.342i)29-s + (−0.5 + 0.866i)31-s i·37-s + (−0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.342 − 0.939i)47-s + (0.5 + 0.866i)49-s + (−0.642 + 0.766i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (−0.5 − 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.342 − 0.939i)17-s + (0.642 − 0.766i)23-s + (0.939 − 0.342i)29-s + (−0.5 + 0.866i)31-s i·37-s + (−0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.342 − 0.939i)47-s + (0.5 + 0.866i)49-s + (−0.642 + 0.766i)53-s + (0.939 + 0.342i)59-s + (−0.766 − 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0457 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0457 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.0457 - 0.998i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ -0.0457 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8741982091 - 0.9151649680i\)
\(L(\frac12)\) \(\approx\) \(0.8741982091 - 0.9151649680i\)
\(L(1)\) \(\approx\) \(1.016914898 - 0.1672934033i\)
\(L(1)\) \(\approx\) \(1.016914898 - 0.1672934033i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.984 - 0.173i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (-0.342 - 0.939i)T \)
53 \( 1 + (-0.642 + 0.766i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.984 + 0.173i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.342 + 0.939i)T \)
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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

−19.705692960677353757164050765384, −19.3614808197245364228726504598, −18.195804894851803561272954286231, −17.688590359907619972993868275988, −17.086735262387748224568466128638, −16.37237910501824701067592386065, −15.356106209173366926017118311116, −14.71348226298360686433219690470, −14.31356617195586005307737211383, −13.17404138817260776280626872681, −12.702001928089160444327065499236, −11.766974722017366984414996033131, −11.11909912485806854205616996186, −10.23879641133050930719865527030, −9.73702150697173363266136453666, −8.72434882947763423919135849818, −7.76218750236965106487508895792, −7.450258007102440621235052461722, −6.4808159275899619006198780537, −5.39095068874838457108034714514, −4.754150209750199525271920493519, −4.06866288178436022130378095141, −2.94903358350978789105826139377, −1.99022159227520769148212883357, −1.22598852484691282266167684891, 0.41969532998251780706029930741, 1.621759951366540125561711977090, 2.654657233272652544408172367250, 3.19487359019961534738709628434, 4.611387391925248138275138496483, 5.07511447349314662128394904644, 5.816933136625295257063902136864, 6.90270361033275755384923284593, 7.61103882586131678691418049067, 8.47785042898584222193176871690, 8.94181334157750277037291613267, 10.07711108647995901505359331598, 10.63425362376986458833310608138, 11.60653419729314211527532769236, 12.03667365375687098106768071011, 12.911312418735546734617084070526, 13.858827071881479250170045985992, 14.35354246564379702473019891034, 15.16642869148754824988178295647, 15.81526327286614983979214693228, 16.67054779681328360897076959005, 17.30544406456130980468119491748, 18.15925925551528621520806376695, 18.67645344765826384179585669546, 19.35593653538230457097504270484

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