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Properties

Label 2-1200-15.8-c1-0-15
Degree $2$
Conductor $1200$
Sign $0.920 + 0.391i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
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  + (−1.41 + i)3-s + (−2.41 + 2.41i)7-s + (1.00 − 2.82i)9-s − 0.828i·11-s + (−3.82 − 3.82i)13-s + (1.82 + 1.82i)17-s − 0.828i·19-s + (1 − 5.82i)21-s + (4.41 − 4.41i)23-s + (1.41 + 5.00i)27-s + 3.65·29-s − 5.65·31-s + (0.828 + 1.17i)33-s + (5.82 − 5.82i)37-s + (9.24 + 1.58i)39-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s + (−0.912 + 0.912i)7-s + (0.333 − 0.942i)9-s − 0.249i·11-s + (−1.06 − 1.06i)13-s + (0.443 + 0.443i)17-s − 0.190i·19-s + (0.218 − 1.27i)21-s + (0.920 − 0.920i)23-s + (0.272 + 0.962i)27-s + 0.679·29-s − 1.01·31-s + (0.144 + 0.203i)33-s + (0.958 − 0.958i)37-s + (1.48 + 0.253i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.920 + 0.391i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8413458289\)
\(L(\frac12)\) \(\approx\) \(0.8413458289\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - i)T \)
5 \( 1 \)
good7 \( 1 + (2.41 - 2.41i)T - 7iT^{2} \)
11 \( 1 + 0.828iT - 11T^{2} \)
13 \( 1 + (3.82 + 3.82i)T + 13iT^{2} \)
17 \( 1 + (-1.82 - 1.82i)T + 17iT^{2} \)
19 \( 1 + 0.828iT - 19T^{2} \)
23 \( 1 + (-4.41 + 4.41i)T - 23iT^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (-5.82 + 5.82i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (0.414 + 0.414i)T + 43iT^{2} \)
47 \( 1 + (-3.58 - 3.58i)T + 47iT^{2} \)
53 \( 1 + (-3 + 3i)T - 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 + (-10.0 + 10.0i)T - 67iT^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (-4.65 - 4.65i)T + 73iT^{2} \)
79 \( 1 - 0.828iT - 79T^{2} \)
83 \( 1 + (-3.24 + 3.24i)T - 83iT^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (1 - i)T - 97iT^{2} \)
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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

−9.675725997888052768439126263248, −9.176866215254083884803871462128, −8.130172147882384497584703182280, −7.02554891175281530016999360864, −6.16206731111710385031210453306, −5.50741524443740044845264631277, −4.72840169249165702237381466792, −3.46248613724721306978545722969, −2.60092252171982828914386518938, −0.51891344622334452814825754826, 0.959123601835265283717060823176, 2.37532715194180432980707942282, 3.73746855740865562908603316494, 4.76281350849780962409744014883, 5.60373077247507310356585937024, 6.82829773912297807694231438333, 7.00445985598341461780265280614, 7.84266179404325849043015135900, 9.213618949871651812320128122800, 9.880451847775986748484055845772

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