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Properties

Label 1-1720-1720.493-r0-0-0
Degree $1$
Conductor $1720$
Sign $0.958 - 0.283i$
Analytic cond. $7.98764$
Root an. cond. $7.98764$
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  + (−0.680 − 0.733i)3-s + (−0.866 − 0.5i)7-s + (−0.0747 + 0.997i)9-s + (0.900 − 0.433i)11-s + (0.930 − 0.365i)13-s + (0.149 + 0.988i)17-s + (−0.0747 − 0.997i)19-s + (0.222 + 0.974i)21-s + (−0.563 + 0.826i)23-s + (0.781 − 0.623i)27-s + (0.733 + 0.680i)29-s + (0.955 − 0.294i)31-s + (−0.930 − 0.365i)33-s + (−0.866 + 0.5i)37-s + (−0.900 − 0.433i)39-s + ⋯
L(s)  = 1  + (−0.680 − 0.733i)3-s + (−0.866 − 0.5i)7-s + (−0.0747 + 0.997i)9-s + (0.900 − 0.433i)11-s + (0.930 − 0.365i)13-s + (0.149 + 0.988i)17-s + (−0.0747 − 0.997i)19-s + (0.222 + 0.974i)21-s + (−0.563 + 0.826i)23-s + (0.781 − 0.623i)27-s + (0.733 + 0.680i)29-s + (0.955 − 0.294i)31-s + (−0.930 − 0.365i)33-s + (−0.866 + 0.5i)37-s + (−0.900 − 0.433i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1720 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1720 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1720\)    =    \(2^{3} \cdot 5 \cdot 43\)
Sign: $0.958 - 0.283i$
Analytic conductor: \(7.98764\)
Root analytic conductor: \(7.98764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1720} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1720,\ (0:\ ),\ 0.958 - 0.283i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.114472621 - 0.1612768770i\)
\(L(\frac12)\) \(\approx\) \(1.114472621 - 0.1612768770i\)
\(L(1)\) \(\approx\) \(0.8438824125 - 0.1756362882i\)
\(L(1)\) \(\approx\) \(0.8438824125 - 0.1756362882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.680 - 0.733i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (0.930 - 0.365i)T \)
17 \( 1 + (0.149 + 0.988i)T \)
19 \( 1 + (-0.0747 - 0.997i)T \)
23 \( 1 + (-0.563 + 0.826i)T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 + (0.955 - 0.294i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (-0.930 - 0.365i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (-0.997 + 0.0747i)T \)
71 \( 1 + (-0.826 + 0.563i)T \)
73 \( 1 + (0.930 - 0.365i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.680 + 0.733i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 + (0.433 + 0.900i)T \)
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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

−20.61448829803216700559335850679, −19.56720720015782524084164962998, −18.79240095494924118832626061001, −18.09760170704760135670614696985, −17.28122606984454499633129402137, −16.48165714396491869015592871767, −15.976707611051390268787249168836, −15.40730710859762948455131031356, −14.393777281744634410977656197829, −13.7747688676859287419733701222, −12.52093812487794079900169575645, −12.05473553670671047592644498486, −11.43363372039129468302024267445, −10.3324020156172426794955984721, −9.85450824262326353382098447459, −9.06430430222283196677831013652, −8.383617615578475725479346480154, −6.92489597795941868502245084756, −6.378281403569180417691361015400, −5.71941465878362922081999653182, −4.703850435952077869188706210526, −3.889986654761536679834726864110, −3.2106134144035641905480214794, −1.9431523709796673860032158441, −0.623850673231914748019398380675, 0.862545788604389094211766703695, 1.53625596938951183913149700720, 2.92389589373332417439214185905, 3.722235501969863822378421509575, 4.71677347639297202739556808063, 5.88299490347983508685479740471, 6.3707070993416799955653553159, 6.9641100530528136253081643939, 8.01442146236177639323973446399, 8.709129152209152835725921579051, 9.79200654710871420206655879677, 10.58063629739228163607509045872, 11.28245554956612535061056059785, 12.02064650641162977628330956200, 12.85195134736464766509026979751, 13.47177407021083307099300381167, 13.9843380121872840514360856202, 15.17824225215956495588750173235, 16.077009209415871495692116261403, 16.56141431103346335111209302481, 17.52826714673120681333199668668, 17.77112752342084923750025410701, 19.01897421325040282863564248016, 19.37319226308712652062624596119, 19.969648073236202823813330588367

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