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Properties

Label 1-6300-6300.2531-r0-0-0
Degree $1$
Conductor $6300$
Sign $0.828 - 0.559i$
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  + (−0.104 − 0.994i)11-s + (0.913 − 0.406i)13-s + (−0.669 + 0.743i)17-s + (0.978 + 0.207i)19-s + (0.913 + 0.406i)23-s + (0.978 − 0.207i)29-s + (−0.309 − 0.951i)31-s + (0.913 − 0.406i)37-s + (−0.913 + 0.406i)41-s + (0.5 + 0.866i)43-s + (0.309 − 0.951i)47-s + (0.978 − 0.207i)53-s + (−0.809 − 0.587i)59-s + (−0.809 + 0.587i)61-s + (−0.309 − 0.951i)67-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)11-s + (0.913 − 0.406i)13-s + (−0.669 + 0.743i)17-s + (0.978 + 0.207i)19-s + (0.913 + 0.406i)23-s + (0.978 − 0.207i)29-s + (−0.309 − 0.951i)31-s + (0.913 − 0.406i)37-s + (−0.913 + 0.406i)41-s + (0.5 + 0.866i)43-s + (0.309 − 0.951i)47-s + (0.978 − 0.207i)53-s + (−0.809 − 0.587i)59-s + (−0.809 + 0.587i)61-s + (−0.309 − 0.951i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.828 - 0.559i$
Analytic conductor: \(29.2570\)
Root analytic conductor: \(29.2570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (2531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6300,\ (0:\ ),\ 0.828 - 0.559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.943392032 - 0.5947319375i\)
\(L(\frac12)\) \(\approx\) \(1.943392032 - 0.5947319375i\)
\(L(1)\) \(\approx\) \(1.177870003 - 0.1137300201i\)
\(L(1)\) \(\approx\) \(1.177870003 - 0.1137300201i\)

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

−17.80622232749662768178386822667, −17.072825694691367932615398107000, −16.292428357128628893559482774, −15.7156817692044782097636295461, −15.23354787064838968253138897673, −14.35398035108394466855056852129, −13.75255200679635938530364835687, −13.21675177099593835489981470180, −12.35638306602269419809990496360, −11.86793541147723708433721842706, −11.05342873381260448536756179081, −10.523213157651532256349436525, −9.68027171086780631500770933111, −9.046314769733220804606735006043, −8.534940163218754709838585375224, −7.5087756378967634046054226739, −6.99624119793174165685383687735, −6.40373309803121384081123059686, −5.42391813213564694949537564942, −4.76505383793044610044976706978, −4.188090604022636025027876662502, −3.176506930157287909545819994758, −2.57504740230815084593074435310, −1.60409916215149372944629403836, −0.84954296265013008645274402149, 0.67025027378051421412429199789, 1.35890348301093291971568997368, 2.41679901060900135625097000701, 3.242780303850525285145819507638, 3.75915492077582874174037563946, 4.67381683477219319766180652963, 5.522268573589989765168070669134, 6.08155273909097575463389525859, 6.70632612758531036474353867336, 7.72556003827572731621105969641, 8.197623531678539453564844213286, 8.926235122275586586082844214745, 9.5338789829109742368674750485, 10.465772726627149843661597950, 11.00079523629302227424596015417, 11.52176026761267710647066546750, 12.32369062905934103146230056415, 13.295287757352731884597022629401, 13.42399573944936879854703807398, 14.229624822386254628025707646471, 15.15208146240614807020533653364, 15.51533079026947633805319564626, 16.34925741922981579995225106782, 16.79588610868286632772826891021, 17.60690427743376822686917540059

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