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Properties

Label 2-1400-1400.69-c0-0-0
Degree $2$
Conductor $1400$
Sign $-0.844 - 0.535i$
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  + (−0.951 + 0.309i)2-s + (1.16 + 1.59i)3-s + (0.809 − 0.587i)4-s + (−0.453 + 0.891i)5-s + (−1.59 − 1.16i)6-s − i·7-s + (−0.587 + 0.809i)8-s + (−0.896 + 2.76i)9-s + (0.156 − 0.987i)10-s + (1.87 + 0.610i)12-s + (0.297 + 0.0966i)13-s + (0.309 + 0.951i)14-s + (−1.95 + 0.309i)15-s + (0.309 − 0.951i)16-s − 2.90i·18-s + (1.14 + 0.831i)19-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (1.16 + 1.59i)3-s + (0.809 − 0.587i)4-s + (−0.453 + 0.891i)5-s + (−1.59 − 1.16i)6-s − i·7-s + (−0.587 + 0.809i)8-s + (−0.896 + 2.76i)9-s + (0.156 − 0.987i)10-s + (1.87 + 0.610i)12-s + (0.297 + 0.0966i)13-s + (0.309 + 0.951i)14-s + (−1.95 + 0.309i)15-s + (0.309 − 0.951i)16-s − 2.90i·18-s + (1.14 + 0.831i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.844 - 0.535i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ -0.844 - 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9241792692\)
\(L(\frac12)\) \(\approx\) \(0.9241792692\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (0.453 - 0.891i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-1.16 - 1.59i)T + (-0.309 + 0.951i)T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.297 - 0.0966i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-1.14 - 0.831i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.550 + 1.69i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.280 - 0.863i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.533 + 0.734i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−9.881019413582762190340790844164, −9.556661750366486258243893492541, −8.423926589139704856385230186150, −7.86112413571080517292845007682, −7.32403840139635459071982454002, −6.09079865289407421331301500256, −4.94057670645329080470428110930, −3.77163447310694315362410557847, −3.32336815792343199554044098938, −2.08441212485587964019482867358, 0.928049194437025365814701901397, 1.99529831607470987842968213744, 2.82190809120925124773434100798, 3.78681556139099661091099261421, 5.58076585976131731478494739108, 6.53408984110519467438104289185, 7.34761262173409223039013108306, 8.120943540111550813296868279445, 8.449563560782696372849036387462, 9.203268257094642702509002289201

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