↑

Properties

Label 1-3520-3520.147-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.894 - 0.446i$
Analytic cond. $16.3468$
Root an. cond. $16.3468$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.522 + 0.852i)3-s + (−0.987 − 0.156i)7-s + (−0.453 + 0.891i)9-s + (−0.996 − 0.0784i)13-s + (−0.309 + 0.951i)17-s + (−0.522 − 0.852i)19-s + (−0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (−0.996 + 0.0784i)27-s + (−0.233 − 0.972i)29-s + (0.309 + 0.951i)31-s + (−0.852 − 0.522i)37-s + (−0.453 − 0.891i)39-s + (−0.156 − 0.987i)41-s + (0.923 − 0.382i)43-s + ⋯
L(s)  = 1  + (0.522 + 0.852i)3-s + (−0.987 − 0.156i)7-s + (−0.453 + 0.891i)9-s + (−0.996 − 0.0784i)13-s + (−0.309 + 0.951i)17-s + (−0.522 − 0.852i)19-s + (−0.382 − 0.923i)21-s + (−0.707 + 0.707i)23-s + (−0.996 + 0.0784i)27-s + (−0.233 − 0.972i)29-s + (0.309 + 0.951i)31-s + (−0.852 − 0.522i)37-s + (−0.453 − 0.891i)39-s + (−0.156 − 0.987i)41-s + (0.923 − 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.894 - 0.446i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.894 - 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8683557404 - 0.2048336196i\)
\(L(\frac12)\) \(\approx\) \(0.8683557404 - 0.2048336196i\)
\(L(1)\) \(\approx\) \(0.8761891106 + 0.2153217500i\)
\(L(1)\) \(\approx\) \(0.8761891106 + 0.2153217500i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.522 + 0.852i)T \)
7 \( 1 + (-0.987 - 0.156i)T \)
13 \( 1 + (-0.996 - 0.0784i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.522 - 0.852i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.233 - 0.972i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.852 - 0.522i)T \)
41 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.649 + 0.760i)T \)
59 \( 1 + (0.522 - 0.852i)T \)
61 \( 1 + (0.760 + 0.649i)T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (-0.453 - 0.891i)T \)
73 \( 1 + (0.156 - 0.987i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (0.760 + 0.649i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.855938389833747319637303231569, −18.27309704171019941626490099365, −17.52435511056682764552134328422, −16.6924069537900590681117055329, −16.12929739527024800757511911764, −15.222118111878840179424150499524, −14.51233177646372879949009585725, −14.00775499715706247706631705077, −13.094533499628590636455739336562, −12.66747732819686588101719416871, −12.030638588090535092356962770, −11.33884523404543119537268615687, −10.09924110597857011414666332340, −9.69569801931366287183670145607, −8.87572154374785058486185173509, −8.17050724410627170576308344093, −7.380056391201932966217858412173, −6.7205824256882036676495024370, −6.17256223499085985840518411957, −5.24308807488598628143467864503, −4.19960625017964832426761021113, −3.32891130562943392980030908671, −2.5768117411700294757788765457, −2.00369241581991182917226621251, −0.76911828145100873770532816248, 0.299083518496192356697227581511, 2.00642336267129848460603815564, 2.56745747914325357284795310369, 3.57842613617646514383993786384, 4.009466731008337032458666918114, 4.95029087458060715244748716965, 5.690282570918049796043727674270, 6.60492293131987855834048839342, 7.36387858031777119316733127210, 8.20876642624123581764562776342, 9.0011989910810220535175424791, 9.527191549360468259557690724084, 10.32109558790582340012264301318, 10.6585449314785163632558714381, 11.74746970360745018208108859780, 12.49532414913694117107806143087, 13.279197299853165399770585816484, 13.858256127138876134558403332193, 14.64590259777470164873525917519, 15.43324559127535762097005586004, 15.7081605332907352925413282949, 16.5933757318274294872480672536, 17.22672171463252861721196898028, 17.769836868408234593153080766935, 19.2024514722239141290405895867

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