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Properties

Label 1-6300-6300.2279-r0-0-0
Degree $1$
Conductor $6300$
Sign $-0.394 - 0.918i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.913 − 0.406i)11-s + (0.104 + 0.994i)13-s + (−0.978 + 0.207i)17-s + (−0.669 − 0.743i)19-s + (0.104 − 0.994i)23-s + (−0.669 + 0.743i)29-s + (−0.309 + 0.951i)31-s + (0.104 + 0.994i)37-s + (0.104 + 0.994i)41-s + (−0.5 − 0.866i)43-s + (−0.309 − 0.951i)47-s + (0.669 − 0.743i)53-s + (−0.809 + 0.587i)59-s + (−0.809 − 0.587i)61-s + (0.309 − 0.951i)67-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)11-s + (0.104 + 0.994i)13-s + (−0.978 + 0.207i)17-s + (−0.669 − 0.743i)19-s + (0.104 − 0.994i)23-s + (−0.669 + 0.743i)29-s + (−0.309 + 0.951i)31-s + (0.104 + 0.994i)37-s + (0.104 + 0.994i)41-s + (−0.5 − 0.866i)43-s + (−0.309 − 0.951i)47-s + (0.669 − 0.743i)53-s + (−0.809 + 0.587i)59-s + (−0.809 − 0.587i)61-s + (0.309 − 0.951i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.394 - 0.918i$
Analytic conductor: \(29.2570\)
Root analytic conductor: \(29.2570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (2279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6300,\ (0:\ ),\ -0.394 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4646273528 - 0.7050610903i\)
\(L(\frac12)\) \(\approx\) \(0.4646273528 - 0.7050610903i\)
\(L(1)\) \(\approx\) \(0.9378146593 - 0.05729223738i\)
\(L(1)\) \(\approx\) \(0.9378146593 - 0.05729223738i\)

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 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.104 + 0.994i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−17.8220925478202659255322022079, −17.11778625022394585057999614992, −16.74763266214753015658692427178, −15.64236221979540896190495484347, −15.34891301946651513940117440581, −14.6071554571908389730908046737, −13.97152309263263493803431044364, −13.120419203017741664047578579259, −12.732574164236020419559347628548, −11.89463841097951509816711732005, −11.24331792836458390467059861439, −10.67424110889612070029489910157, −9.78730552847757935759647345300, −9.29209805193289006421500540595, −8.5762418216239034131173546165, −7.714047425764668455473666694254, −7.27072821702281155376344936492, −6.24133654468393320134611709574, −5.86651962911790342055940116768, −4.941079941564034573570088597269, −4.06974515297206708832701466098, −3.658572811896935613618012529485, −2.56711865519834773235940007082, −1.89243389223473626116442444117, −0.995690117071456248082953037875, 0.21792226216458327959665593513, 1.455123978388925963236215683630, 2.02769569213298257327847084778, 3.01114027948742088473618459815, 3.79529898960161597970392478124, 4.509413456005488440046582409834, 5.06529270460884540695149984631, 6.24984343115774979180082195057, 6.61560205312379556211276710002, 7.153843012009496528764719949712, 8.3431453900791479784028521433, 8.82133582466121243874799658130, 9.25774325069106608544026472874, 10.22574919442933277545362502665, 10.958792081477089133217738342491, 11.42944847780206766561835035240, 12.14246172972457009736892763906, 12.86908482674891561919889770534, 13.577853486384862760502633717240, 14.11146993353356646911612900784, 14.9133130068445125798751232284, 15.29462769511817540204697048860, 16.41247543602652544018095662442, 16.631390373560939866473466847371, 17.32232575334546355504483576944

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