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Properties

Label 2-6300-105.59-c1-0-13
Degree $2$
Conductor $6300$
Sign $0.0458 - 0.998i$
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  + (2.08 + 1.63i)7-s + (−5.10 − 2.94i)11-s − 4.65·13-s + (1.82 + 1.05i)17-s + (4.55 − 2.62i)19-s + (3.91 + 6.78i)23-s + 0.273i·29-s + (−4.27 − 2.46i)31-s + (5.05 − 2.91i)37-s − 2.26·41-s − 8.63i·43-s + (−4.31 + 2.49i)47-s + (1.67 + 6.79i)49-s + (−2.81 + 4.87i)53-s + (−2.33 + 4.05i)59-s + ⋯
L(s)  = 1  + (0.787 + 0.616i)7-s + (−1.54 − 0.889i)11-s − 1.29·13-s + (0.441 + 0.254i)17-s + (1.04 − 0.603i)19-s + (0.816 + 1.41i)23-s + 0.0508i·29-s + (−0.767 − 0.442i)31-s + (0.831 − 0.479i)37-s − 0.353·41-s − 1.31i·43-s + (−0.629 + 0.363i)47-s + (0.239 + 0.970i)49-s + (−0.386 + 0.669i)53-s + (−0.304 + 0.527i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0458 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0458 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.0458 - 0.998i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ 0.0458 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.324127041\)
\(L(\frac12)\) \(\approx\) \(1.324127041\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.08 - 1.63i)T \)
good11 \( 1 + (5.10 + 2.94i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.65T + 13T^{2} \)
17 \( 1 + (-1.82 - 1.05i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.55 + 2.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.91 - 6.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.273iT - 29T^{2} \)
31 \( 1 + (4.27 + 2.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.05 + 2.91i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.26T + 41T^{2} \)
43 \( 1 + 8.63iT - 43T^{2} \)
47 \( 1 + (4.31 - 2.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.81 - 4.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.33 - 4.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.21 + 4.74i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.46 - 3.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + (-0.370 + 0.642i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.47 - 2.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.84iT - 83T^{2} \)
89 \( 1 + (-3.40 - 5.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.90T + 97T^{2} \)
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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

−7.987230941371087075867680311861, −7.66887442795736599415689753466, −7.02080855366228742100071197247, −5.75731095872453793467324804897, −5.28506411243509172817439802876, −4.99251843838510318276076787126, −3.71425892860565883832259989457, −2.82375437155051144413083618182, −2.26198154324852986091928995249, −0.988014832001577775518277469785, 0.36854564085492302465812385818, 1.64799790075879068463947362890, 2.54503109117055617279874395561, 3.28644685137143555914439788518, 4.57621806983434632038508433889, 4.89599509921171890025870626885, 5.45492374243375099331791591083, 6.65964292667705844509790187658, 7.34143888976673777000105465707, 7.79185973248153953179079518638

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