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Properties

Label 2-6300-21.17-c1-0-17
Degree $2$
Conductor $6300$
Sign $0.999 - 0.0243i$
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  + (−1.29 − 2.30i)7-s + (−1.33 − 0.768i)11-s − 0.219i·13-s + (0.713 − 1.23i)17-s + (−3.80 + 2.19i)19-s + (−0.673 + 0.389i)23-s + 6.82i·29-s + (5.32 + 3.07i)31-s + (2.99 + 5.19i)37-s − 8.90·41-s − 1.91·43-s + (5.60 + 9.70i)47-s + (−3.62 + 5.98i)49-s + (4.18 + 2.41i)53-s + (0.336 − 0.582i)59-s + ⋯
L(s)  = 1  + (−0.490 − 0.871i)7-s + (−0.401 − 0.231i)11-s − 0.0608i·13-s + (0.172 − 0.299i)17-s + (−0.872 + 0.503i)19-s + (−0.140 + 0.0811i)23-s + 1.26i·29-s + (0.956 + 0.552i)31-s + (0.492 + 0.853i)37-s − 1.39·41-s − 0.292·43-s + (0.817 + 1.41i)47-s + (−0.518 + 0.855i)49-s + (0.574 + 0.331i)53-s + (0.0437 − 0.0758i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.999 - 0.0243i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (4301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ 0.999 - 0.0243i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.479274996\)
\(L(\frac12)\) \(\approx\) \(1.479274996\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 + 2.30i)T \)
good11 \( 1 + (1.33 + 0.768i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.219iT - 13T^{2} \)
17 \( 1 + (-0.713 + 1.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.80 - 2.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.673 - 0.389i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.82iT - 29T^{2} \)
31 \( 1 + (-5.32 - 3.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.99 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.90T + 41T^{2} \)
43 \( 1 + 1.91T + 43T^{2} \)
47 \( 1 + (-5.60 - 9.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.18 - 2.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.336 + 0.582i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.64 + 5.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.65 + 9.78i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + (1.88 + 1.09i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.19 + 5.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.09T + 83T^{2} \)
89 \( 1 + (-2.97 - 5.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.7iT - 97T^{2} \)
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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

−8.045316471348587623235933313779, −7.31235199708929248191474972765, −6.62290696892860848501952110115, −6.07419137165348865738741243130, −5.07331739909457531871898347982, −4.46464376363594717681463285337, −3.51350700969224775283842567561, −2.96392243907489154058744596315, −1.76576306452209620335062193956, −0.68193502591586643432370889422, 0.56195921557517308292363659846, 2.15212639862435527424749403059, 2.52574583326005004876145772741, 3.65737782756810352240011004840, 4.37121246643170749038605223624, 5.31236540424683469184128419614, 5.87833676679504510389507180547, 6.61364659594462110402256866891, 7.23515154453276075437223502110, 8.317087028104678643426840207016

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