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Properties

Label 2-1400-56.45-c0-0-0
Degree $2$
Conductor $1400$
Sign $-0.605 - 0.795i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)7-s − 0.999·8-s + (0.5 + 0.866i)9-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)23-s − 0.999·28-s + (1.5 + 0.866i)31-s + (0.499 − 0.866i)32-s − 1.73i·34-s − 0.999·36-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)7-s − 0.999·8-s + (0.5 + 0.866i)9-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.5 + 0.866i)23-s − 0.999·28-s + (1.5 + 0.866i)31-s + (0.499 − 0.866i)32-s − 1.73i·34-s − 0.999·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ -0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.327724631\)
\(L(\frac12)\) \(\approx\) \(1.327724631\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + 1.73iT - T^{2} \)
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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

−9.819892793658799178902341034078, −8.943008756550614501453968928576, −8.396026262808958145846998167879, −7.49644272216237756518809882823, −6.83341114731026566192459680126, −5.87507935106721267083751221869, −4.90853706078785430990466233289, −4.58511237240076559861826811341, −3.13326258275683580224938435960, −2.08030175307023728791051325317, 1.00561290690679019429405129794, 2.22311493512140578088754613390, 3.49245235835361237503992114451, 4.33494797215586043126524890271, 4.83766775701998998320063301831, 6.34757648034035299142123961903, 6.65274252191565286689560003681, 8.040552502300954081511740304015, 8.806630611356733707179497744373, 9.755735906599339843976681354806

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