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Properties

Label 2-2800-112.51-c0-0-3
Degree $2$
Conductor $2800$
Sign $0.557 - 0.830i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
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  + (0.866 + 0.5i)2-s + (1.36 + 0.366i)3-s + (0.499 + 0.866i)4-s + (0.999 + i)6-s + (0.5 − 0.866i)7-s + 0.999i·8-s + (0.866 + 0.5i)9-s + (0.366 + 1.36i)12-s + (0.866 − 0.499i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (−0.366 − 1.36i)19-s + (1 − 0.999i)21-s + (−0.5 + 0.866i)23-s + (−0.366 + 1.36i)24-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (1.36 + 0.366i)3-s + (0.499 + 0.866i)4-s + (0.999 + i)6-s + (0.5 − 0.866i)7-s + 0.999i·8-s + (0.866 + 0.5i)9-s + (0.366 + 1.36i)12-s + (0.866 − 0.499i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.499 + 0.866i)18-s + (−0.366 − 1.36i)19-s + (1 − 0.999i)21-s + (−0.5 + 0.866i)23-s + (−0.366 + 1.36i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 0.557 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.185050510\)
\(L(\frac12)\) \(\approx\) \(3.185050510\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.018826513105410812784067520038, −8.170305339629014645875345231745, −7.56977121785695218628616730805, −7.03842978114309470764256585567, −6.07186820625665122916405694442, −4.80522373725489614198043773439, −4.48821190360122387413952157273, −3.49396950574995535034128412387, −2.87539626680324272731590646892, −1.85844084017632338425147370706, 1.88601150939364047850047652197, 2.04012079620715939811923799099, 3.18531580740068392695133761057, 3.88897427516696178715612281313, 4.76120927301123882091221626267, 5.84700223552903920712179513646, 6.34951477895274044193521492958, 7.55102928252001869596473556251, 8.103310086249247929060653280944, 8.895235840429188902704498728757

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