↑

Properties

Label 1-6300-6300.587-r0-0-0
Degree $1$
Conductor $6300$
Sign $0.975 + 0.221i$
Analytic cond. $29.2570$
Root an. cond. $29.2570$
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  + (0.669 − 0.743i)11-s + (−0.743 + 0.669i)13-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.207 + 0.978i)23-s + (0.913 + 0.406i)29-s + (0.913 − 0.406i)31-s + (0.951 + 0.309i)37-s + (0.669 + 0.743i)41-s + (0.866 − 0.5i)43-s + (0.406 − 0.913i)47-s + (−0.587 − 0.809i)53-s + (0.669 + 0.743i)59-s + (−0.669 + 0.743i)61-s + (−0.406 − 0.913i)67-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)11-s + (−0.743 + 0.669i)13-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.207 + 0.978i)23-s + (0.913 + 0.406i)29-s + (0.913 − 0.406i)31-s + (0.951 + 0.309i)37-s + (0.669 + 0.743i)41-s + (0.866 − 0.5i)43-s + (0.406 − 0.913i)47-s + (−0.587 − 0.809i)53-s + (0.669 + 0.743i)59-s + (−0.669 + 0.743i)61-s + (−0.406 − 0.913i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.975 + 0.221i$
Analytic conductor: \(29.2570\)
Root analytic conductor: \(29.2570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6300,\ (0:\ ),\ 0.975 + 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.031491915 + 0.2278683284i\)
\(L(\frac12)\) \(\approx\) \(2.031491915 + 0.2278683284i\)
\(L(1)\) \(\approx\) \(1.185358278 + 0.02313162983i\)
\(L(1)\) \(\approx\) \(1.185358278 + 0.02313162983i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (-0.743 + 0.669i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.207 + 0.978i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.406 - 0.913i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (-0.406 - 0.913i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.994 - 0.104i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.406 - 0.913i)T \)
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

−17.582899154824565486127545285945, −17.139639366362044242944067798, −16.237310765314796658446721734079, −15.681738990837046882171498189417, −14.89276679149167765920101774126, −14.44348486603947139179543992687, −13.804094250195875029481370209203, −12.86145796381954787659130759917, −12.31564915253734653175061804766, −11.93080025965324921903803838028, −10.9070031001565462309327040464, −10.34671329115102632198065462135, −9.636118015988050420948036134370, −9.131721728044093961630099684771, −8.11807702909150886886057797780, −7.681681950699946714530963114280, −6.867240614359337819665322869259, −6.18608268466675325120276234374, −5.46755067043944572128672313113, −4.53227980959674879475908601468, −4.186838474875200914345463906416, −2.98280927772635314749684280446, −2.56218740657667719458481610957, −1.46795084058430706487999976079, −0.69597547212162545364342758186, 0.81661906433569527564142660394, 1.47016109469767159349542567226, 2.584623685138511974701921243675, 3.17209776593780787195970170820, 4.03370606620103107601860534547, 4.71625895337940960816176381737, 5.57971315545377540605754699045, 6.12730592995456325549118198367, 7.0265215499481904989665160461, 7.56526453594271182068747076039, 8.32239535305620808164925952988, 9.1427056464708650130692196609, 9.68928402728641701517219492742, 10.23921501021950392158450584116, 11.29619610388493154337012806354, 11.79727579384696786200790581342, 12.14681869873518334688041272722, 13.19871300991887386116009587564, 13.89466490292569909036864742828, 14.26129650858842155090902277447, 14.944876977586817656180815051334, 15.88776292321046378236582213163, 16.33697756201338432269089304981, 16.92848039524635883998827294330, 17.61357278294196802197093102735

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