Properties

Label 2-1400-1400.909-c0-0-3
Degree $2$
Conductor $1400$
Sign $-0.684 + 0.728i$
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  + (0.587 − 0.809i)2-s + (0.863 − 0.280i)3-s + (−0.309 − 0.951i)4-s + (−0.156 − 0.987i)5-s + (0.280 − 0.863i)6-s i·7-s + (−0.951 − 0.309i)8-s + (−0.142 + 0.103i)9-s + (−0.891 − 0.453i)10-s + (−0.533 − 0.734i)12-s + (1.04 + 1.44i)13-s + (−0.809 − 0.587i)14-s + (−0.412 − 0.809i)15-s + (−0.809 + 0.587i)16-s + 0.175i·18-s + (−0.437 + 1.34i)19-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (0.863 − 0.280i)3-s + (−0.309 − 0.951i)4-s + (−0.156 − 0.987i)5-s + (0.280 − 0.863i)6-s i·7-s + (−0.951 − 0.309i)8-s + (−0.142 + 0.103i)9-s + (−0.891 − 0.453i)10-s + (−0.533 − 0.734i)12-s + (1.04 + 1.44i)13-s + (−0.809 − 0.587i)14-s + (−0.412 − 0.809i)15-s + (−0.809 + 0.587i)16-s + 0.175i·18-s + (−0.437 + 1.34i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ -0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.743406245\)
\(L(\frac12)\) \(\approx\) \(1.743406245\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (0.156 + 0.987i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-0.863 + 0.280i)T + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-1.04 - 1.44i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
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 \( 1 + (-1.59 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.253 + 0.183i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.297 - 0.0966i)T + (0.809 + 0.587i)T^{2} \)
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

−9.393359129947616975970819074507, −8.689656797300675080027467992540, −8.198141441871708149736001086826, −6.96714884545689417967051739348, −6.09088616887626426564434485704, −4.91405983871554474482913259079, −4.12361852302141855469205038704, −3.51526723981625908947369938262, −2.13390926212324365417884007073, −1.23065065796331416808026768510, 2.63828586661358771614611818293, 3.10257271838483219606383702998, 3.88062347221887962136328289255, 5.27277020796511396516556789809, 5.89440184407941572637292602826, 6.78019002166400121916917146704, 7.65761748104023727048479332378, 8.425871026595789518066969352425, 8.951190813012240682821858101641, 9.764355883517378783119242513692

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