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Properties

Label 2-6300-21.5-c1-0-38
Degree $2$
Conductor $6300$
Sign $0.617 + 0.786i$
Analytic cond. $50.3057$
Root an. cond. $7.09265$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.60 − 0.464i)7-s + (3.45 − 1.99i)11-s + 2.37i·13-s + (−1.11 − 1.93i)17-s + (−2.78 − 1.61i)19-s + (−2.87 − 1.65i)23-s + 4.57i·29-s + (6.29 − 3.63i)31-s + (2.07 − 3.59i)37-s − 0.917·41-s + 1.34·43-s + (1.17 − 2.03i)47-s + (6.56 − 2.42i)49-s + (4.46 − 2.57i)53-s + (−2.20 − 3.81i)59-s + ⋯
L(s)  = 1  + (0.984 − 0.175i)7-s + (1.04 − 0.601i)11-s + 0.659i·13-s + (−0.270 − 0.468i)17-s + (−0.640 − 0.369i)19-s + (−0.599 − 0.345i)23-s + 0.849i·29-s + (1.13 − 0.652i)31-s + (0.340 − 0.590i)37-s − 0.143·41-s + 0.205·43-s + (0.171 − 0.297i)47-s + (0.938 − 0.345i)49-s + (0.613 − 0.354i)53-s + (−0.286 − 0.496i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ 0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.301364764\)
\(L(\frac12)\) \(\approx\) \(2.301364764\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.60 + 0.464i)T \)
good11 \( 1 + (-3.45 + 1.99i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.37iT - 13T^{2} \)
17 \( 1 + (1.11 + 1.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.78 + 1.61i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.87 + 1.65i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.57iT - 29T^{2} \)
31 \( 1 + (-6.29 + 3.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.07 + 3.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.917T + 41T^{2} \)
43 \( 1 - 1.34T + 43T^{2} \)
47 \( 1 + (-1.17 + 2.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.46 + 2.57i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.20 + 3.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.38 - 2.53i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.01 + 1.76i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 16.6iT - 71T^{2} \)
73 \( 1 + (-1.34 + 0.776i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.89 - 3.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.917T + 83T^{2} \)
89 \( 1 + (-4.05 + 7.01i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.030176848471879505961530821790, −7.17554387594863219767323986174, −6.55687467920135266662062495664, −5.89981467851593025362625730030, −4.92099875997702768294321811235, −4.33346693085112689609136000391, −3.67414928546953396062315243655, −2.50209869216533017907674978389, −1.69283757217007666987481483009, −0.64731533155585823472993219745, 1.08039099759661752425869929772, 1.89728842837373655153104311858, 2.78324031203511159473696263559, 4.02702575501034069885986582982, 4.34985718005708573292089418125, 5.30310567833256957798348819865, 6.03059038183533507688169986472, 6.69882227080013525729359417892, 7.51453063796575424494717961275, 8.256830201127162267042515566284

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