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Properties

Label 1-2541-2541.782-r0-0-0
Degree $1$
Conductor $2541$
Sign $-0.988 - 0.149i$
Analytic cond. $11.8003$
Root an. cond. $11.8003$
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  + (−0.0475 + 0.998i)2-s + (−0.995 − 0.0950i)4-s + (−0.786 + 0.618i)5-s + (0.142 − 0.989i)8-s + (−0.580 − 0.814i)10-s + (−0.415 − 0.909i)13-s + (0.981 + 0.189i)16-s + (0.235 − 0.971i)17-s + (−0.235 − 0.971i)19-s + (0.841 − 0.540i)20-s + (−0.981 − 0.189i)23-s + (0.235 − 0.971i)25-s + (0.928 − 0.371i)26-s + (0.959 + 0.281i)29-s + (−0.580 − 0.814i)31-s + (−0.235 + 0.971i)32-s + ⋯
L(s)  = 1  + (−0.0475 + 0.998i)2-s + (−0.995 − 0.0950i)4-s + (−0.786 + 0.618i)5-s + (0.142 − 0.989i)8-s + (−0.580 − 0.814i)10-s + (−0.415 − 0.909i)13-s + (0.981 + 0.189i)16-s + (0.235 − 0.971i)17-s + (−0.235 − 0.971i)19-s + (0.841 − 0.540i)20-s + (−0.981 − 0.189i)23-s + (0.235 − 0.971i)25-s + (0.928 − 0.371i)26-s + (0.959 + 0.281i)29-s + (−0.580 − 0.814i)31-s + (−0.235 + 0.971i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $-0.988 - 0.149i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(11.8003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (782, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2541,\ (0:\ ),\ -0.988 - 0.149i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02574573114 + 0.3421873743i\)
\(L(\frac12)\) \(\approx\) \(0.02574573114 + 0.3421873743i\)
\(L(1)\) \(\approx\) \(0.5947194400 + 0.3116195344i\)
\(L(1)\) \(\approx\) \(0.5947194400 + 0.3116195344i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0475 + 0.998i)T \)
5 \( 1 + (-0.786 + 0.618i)T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (0.235 - 0.971i)T \)
19 \( 1 + (-0.235 - 0.971i)T \)
23 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (-0.580 - 0.814i)T \)
37 \( 1 + (-0.995 + 0.0950i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.888 - 0.458i)T \)
53 \( 1 + (-0.981 + 0.189i)T \)
59 \( 1 + (0.0475 + 0.998i)T \)
61 \( 1 + (0.888 + 0.458i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (0.327 + 0.945i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−19.35011709188188874081484581816, −18.619778030723588976388108217260, −17.658798726572548608698955784552, −17.03793977336644016693325949182, −16.303652844716084631451603950898, −15.564989524497143781270153235250, −14.432361174579139343731636169808, −14.12398567605934405450051532251, −13.0264201742798770184997906581, −12.274801773767663450238668263039, −12.0928979055618768332870236756, −11.16284236349360238258183414388, −10.42006574893587274905746662322, −9.689803672637758123155052255967, −8.85609494078017253240288861978, −8.24762188853804328302767652002, −7.60274944549834171762697668192, −6.42000415399018426674436616638, −5.38869227227556277471037929239, −4.61037889719141952703756672220, −3.88736698181007496753144027448, −3.36001824100184284719850542883, −2.00726757537090846448403246112, −1.492301192504576214191859933942, −0.145934187093730732118212641505, 0.87301500839016241396289158673, 2.56231652240495053533384202245, 3.29984943592843749979168029902, 4.2776794250204275944672814770, 4.93136501869165863565525732282, 5.82524454124342328088893385730, 6.69796026857211109518657231477, 7.29688281505428464219326701572, 7.958762034482138550399098287974, 8.582930035831716571435324301565, 9.6080893134578466895572078133, 10.21459056555263778285377131273, 11.11274606004752877950550162053, 11.94821472103702355414331377518, 12.742559481559799653139554123208, 13.510029183051864325496143769020, 14.405131713266654831938395213335, 14.7807102582469144633139226393, 15.73151719995909707882411776015, 15.94173205169143178933173215733, 16.86522259823857697472000676718, 17.8539889050505602541151737375, 18.06958046034159203504143669071, 18.97775285905289239874783674704, 19.66152417557494686336132095450

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