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Properties

Label 2-1400-1400.69-c0-0-2
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 + (−0.183 − 0.253i)3-s + (0.809 − 0.587i)4-s + (0.891 + 0.453i)5-s + (−0.253 − 0.183i)6-s + i·7-s + (0.587 − 0.809i)8-s + (0.278 − 0.857i)9-s + (0.987 + 0.156i)10-s + (−0.297 − 0.0966i)12-s + (−1.87 − 0.610i)13-s + (0.309 + 0.951i)14-s + (−0.0489 − 0.309i)15-s + (0.309 − 0.951i)16-s − 0.902i·18-s + (1.14 + 0.831i)19-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (−0.183 − 0.253i)3-s + (0.809 − 0.587i)4-s + (0.891 + 0.453i)5-s + (−0.253 − 0.183i)6-s + i·7-s + (0.587 − 0.809i)8-s + (0.278 − 0.857i)9-s + (0.987 + 0.156i)10-s + (−0.297 − 0.0966i)12-s + (−1.87 − 0.610i)13-s + (0.309 + 0.951i)14-s + (−0.0489 − 0.309i)15-s + (0.309 − 0.951i)16-s − 0.902i·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

\(L(\frac{1}{2})\) \(\approx\) \(2.028117649\)
\(L(\frac12)\) \(\approx\) \(2.028117649\)
\(L(1)\) 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 \( 1 + (-0.891 - 0.453i)T \)
7 \( 1 - iT \)
good3 \( 1 + (0.183 + 0.253i)T + (-0.309 + 0.951i)T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (1.87 + 0.610i)T + (0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.280 - 0.863i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.550 + 1.69i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (-1.04 + 1.44i)T + (-0.309 - 0.951i)T^{2} \)
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

−9.841967956315691320750826429130, −9.288930367087278877247040517160, −7.74627494988425798573129731496, −7.05964090527773464832662590593, −6.06071584229649298991441730293, −5.64930937808745093758752524026, −4.81829150454949942754530871528, −3.42797768097691663014957161764, −2.62240078266992149702370769786, −1.68188351044133965966852190134, 1.82697230194712186374138068627, 2.78154185942145471756386624848, 4.26537246681656822699010508921, 4.78715033374119521096455018243, 5.42224433508523688541927891746, 6.50532817333972885169857795441, 7.33542967000014132527383437314, 7.81904520006037349354209749095, 9.159004775533359671127696639297, 10.03528197441663984695280810239

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