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Properties

Label 2-2800-112.51-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.3071540221\)
\(L(\frac12)\) \(\approx\) \(0.3071540221\)
\(L(1)\) 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

−8.941171075143525831292744625518, −8.685313135201883843996551276326, −7.48818225149968974220628459549, −6.79384035069917262297426457039, −6.20883478404075579934730526601, −5.44973803210816692214986738967, −4.45427469354180078289241704104, −3.21462966946900686649190363239, −2.28475168576084097572108406813, −1.04432962572594809286626376168, 0.38165408112998811693644092398, 1.62996169475956922490176160504, 3.28794493411253247085022510553, 4.38051288462428380002744058779, 5.33068626289643752685527041076, 5.83741214279364952199422858003, 6.60883064968917911115024257352, 7.30330964703426471057498427167, 7.909511504165336096230602654693, 9.036313810237062076136303691635

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