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Properties

Label 2-1400-1400.461-c0-0-1
Degree $2$
Conductor $1400$
Sign $0.728 - 0.684i$
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  + (−0.809 + 0.587i)2-s + (−0.190 + 0.587i)3-s + (0.309 − 0.951i)4-s + (0.809 − 0.587i)5-s + (−0.190 − 0.587i)6-s + 7-s + (0.309 + 0.951i)8-s + (0.5 + 0.363i)9-s + (−0.309 + 0.951i)10-s + (0.5 + 0.363i)12-s + (0.5 + 0.363i)13-s + (−0.809 + 0.587i)14-s + (0.190 + 0.587i)15-s + (−0.809 − 0.587i)16-s − 0.618·18-s + (−0.618 − 1.90i)19-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.190 + 0.587i)3-s + (0.309 − 0.951i)4-s + (0.809 − 0.587i)5-s + (−0.190 − 0.587i)6-s + 7-s + (0.309 + 0.951i)8-s + (0.5 + 0.363i)9-s + (−0.309 + 0.951i)10-s + (0.5 + 0.363i)12-s + (0.5 + 0.363i)13-s + (−0.809 + 0.587i)14-s + (0.190 + 0.587i)15-s + (−0.809 − 0.587i)16-s − 0.618·18-s + (−0.618 − 1.90i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9423269455\)
\(L(\frac12)\) \(\approx\) \(0.9423269455\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 - T \)
good3 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−9.653644536506478439216458809159, −9.071959407010177717738847384378, −8.426939850420612471596641734860, −7.52993604948448659404770427012, −6.63469588782169186030921665953, −5.69763042689353483256138357909, −4.90216729498878839613961551833, −4.38394864580764029181426475342, −2.30713421645267976560776689669, −1.36166945667014705849167415532, 1.43905388031223434604736478736, 2.00286324226684587564999816918, 3.34593507378023861084359138157, 4.37348790438039167033285866280, 5.87374430171563989399850384813, 6.43520394811542215432462433999, 7.49870999223209872174841032957, 7.992385080686473461120322207889, 8.896973722523973277599562486752, 9.811875276469020492733466747299

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