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Properties

Label 1-6300-6300.2659-r0-0-0
Degree $1$
Conductor $6300$
Sign $0.889 - 0.457i$
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  + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.669 − 0.743i)23-s + (−0.104 − 0.994i)29-s + (−0.104 + 0.994i)31-s + (−0.309 + 0.951i)37-s + (0.978 − 0.207i)41-s + (−0.5 + 0.866i)43-s + (0.104 + 0.994i)47-s + (0.809 − 0.587i)53-s + (−0.978 + 0.207i)59-s + (0.978 + 0.207i)61-s + (−0.104 + 0.994i)67-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.669 − 0.743i)23-s + (−0.104 − 0.994i)29-s + (−0.104 + 0.994i)31-s + (−0.309 + 0.951i)37-s + (0.978 − 0.207i)41-s + (−0.5 + 0.866i)43-s + (0.104 + 0.994i)47-s + (0.809 − 0.587i)53-s + (−0.978 + 0.207i)59-s + (0.978 + 0.207i)61-s + (−0.104 + 0.994i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.889 - 0.457i$
Analytic conductor: \(29.2570\)
Root analytic conductor: \(29.2570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (2659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6300,\ (0:\ ),\ 0.889 - 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.476076435 - 0.3571004720i\)
\(L(\frac12)\) \(\approx\) \(1.476076435 - 0.3571004720i\)
\(L(1)\) \(\approx\) \(1.019973955 - 0.04127447692i\)
\(L(1)\) \(\approx\) \(1.019973955 - 0.04127447692i\)

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

−17.46998592001501836737258314103, −17.067315523173525531749604700263, −16.569341837158791740090700831796, −15.63417056769415056367571816243, −14.93090719386114158557824446617, −14.60692278434543354901118902454, −13.77131380479455943826418173772, −13.05461289518689299602097189478, −12.42453240921997067189345938366, −11.84928160370491500018062425689, −11.00677897599611575878364917828, −10.5533509081001295037921909126, −9.628149788456706769607947903141, −9.07702743176380749747297396459, −8.467109851171918594573820087961, −7.57411385450256941385021993715, −6.97685533732873861153335731962, −6.26205767861102198770758773057, −5.555134383479210545812089203853, −4.762575772577626028213665877709, −3.95605429525948829782608184041, −3.44019965111056354194264707924, −2.30535681732029683472506327160, −1.79265020615904155814040588604, −0.69027325989302245887304340776, 0.52598404022463797500417027287, 1.55642002279680462157296773997, 2.43711339159577957412426701420, 2.97643220822054029000402718576, 4.21144673826885578823892327573, 4.50249316305403192528483876807, 5.304054061759581145762013191068, 6.357101349537434480277870987470, 6.81271353064785318283511107901, 7.38196484842652475575901690575, 8.433966035750378363493762006287, 8.92065321340602070374935608500, 9.64365663609616599229671565243, 10.20255957043237056034260204371, 11.27416774520251805836186029221, 11.45166920467231358345980138134, 12.50964249512659873083293219441, 12.797284477659019561417633651342, 13.817198214705658810201572780916, 14.28335496589290950529127343322, 15.03967342941178906802283632481, 15.46839459153567095260995698169, 16.41812598623574256070008647347, 16.9689712924030286926955175813, 17.50960841643327873502471055016

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