Properties

Label 2-1400-1400.69-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.890179394\)
\(L(\frac12)\) \(\approx\) \(1.890179394\)
\(L(1)\) 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

−9.654660614105021222094412605232, −8.804721286273900514184579052272, −8.319191556700376378328819773381, −6.95992286148179343730564677011, −6.26219518719346946245889474827, −5.47929252920317047447232346336, −4.20362938791200721666884555035, −3.94638273737825084054930926745, −2.82821257875307818588889432457, −1.45264672726051789130269908628, 1.72390749674610302089145511011, 3.13005740740165338049053689735, 3.95480635087920498343441836677, 4.45849986981699923236418040231, 5.83757631590870573473678956610, 6.54161797383864509301113536233, 7.36982483989745460662721807094, 8.131056345861204868583311737431, 8.393198244942516690358709597942, 10.37622646929521684315276211271

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