Properties

Label 2-1400-1400.461-c0-0-2
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 + (−0.587 + 1.80i)3-s + (0.309 − 0.951i)4-s + (−0.587 − 0.809i)5-s + (0.587 + 1.80i)6-s − 7-s + (−0.309 − 0.951i)8-s + (−2.11 − 1.53i)9-s + (−0.951 − 0.309i)10-s + (1.53 + 1.11i)12-s + (−1.53 − 1.11i)13-s + (−0.809 + 0.587i)14-s + (1.80 − 0.587i)15-s + (−0.809 − 0.587i)16-s − 2.61·18-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.587 + 1.80i)3-s + (0.309 − 0.951i)4-s + (−0.587 − 0.809i)5-s + (0.587 + 1.80i)6-s − 7-s + (−0.309 − 0.951i)8-s + (−2.11 − 1.53i)9-s + (−0.951 − 0.309i)10-s + (1.53 + 1.11i)12-s + (−1.53 − 1.11i)13-s + (−0.809 + 0.587i)14-s + (1.80 − 0.587i)15-s + (−0.809 − 0.587i)16-s − 2.61·18-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

\(L(\frac{1}{2})\) \(\approx\) \(0.4207377035\)
\(L(\frac12)\) \(\approx\) \(0.4207377035\)
\(L(1)\) 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 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + T \)
good3 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.602565243220173100161443657358, −9.277510884324734512567284253942, −7.966693095700934879688907089237, −6.62229065319014939290435191279, −5.53231800966955359884343061761, −5.17308502305475015503440152735, −4.31235269720788028879397909266, −3.59567777942205648497474576155, −2.80773861137065744452294652644, −0.26003480689122438151243226334, 2.22580581041697133799386198089, 2.87030431347553933941278850483, 4.19190532648003138063654156884, 5.38039590748610933283239612216, 6.33663748713288865816398516465, 6.74590527902205714897938412729, 7.32097094007571088730854718975, 7.86611207059871618923586164257, 8.930164656157059179863126857676, 10.24301713716002207965098355252

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