Properties

Label 2-1400-280.229-c0-0-0
Degree $2$
Conductor $1400$
Sign $0.897 - 0.441i$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
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 + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.5 + 0.866i)9-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (0.866 − 0.499i)18-s + (0.866 + 0.5i)23-s − 0.999i·28-s + (1.5 − 0.866i)31-s + (0.866 − 0.499i)32-s − 1.73i·34-s − 0.999·36-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.5 + 0.866i)9-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + (0.866 − 0.499i)18-s + (0.866 + 0.5i)23-s − 0.999i·28-s + (1.5 − 0.866i)31-s + (0.866 − 0.499i)32-s − 1.73i·34-s − 0.999·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.897 - 0.441i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.897 - 0.441i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6079343496\)
\(L(\frac12)\) \(\approx\) \(0.6079343496\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.73T + T^{2} \)
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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

−9.902843052736704120766080394402, −9.155518401249681928513130926407, −8.064151167198344532055962641848, −7.81590633233448395869169405841, −6.63532427232161056470101490968, −5.95474826851794784163257743483, −4.56140998207264830748760057485, −3.47160226851284291616910371761, −2.69880606878090054973636048477, −1.31612938331364370024605107047, 0.74138255448435302885048051543, 2.54372587052313966033813673154, 3.35286231215934185708487548345, 5.02172893201302514963705171709, 5.72505500947506354828497585868, 6.69196782425426827306123981798, 7.07467259178988899058909605317, 8.314252624983341951786152656157, 8.897192729319005586114327657601, 9.609269781558876904826752804907

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