Properties

Label 2-1400-56.11-c0-0-0
Degree $2$
Conductor $1400$
Sign $0.832 + 0.553i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s i·7-s + 0.999i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s i·13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s + (−0.5 − 0.866i)19-s + 0.999i·22-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + (−0.866 − 0.499i)28-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s i·7-s + 0.999i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s i·13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.499i)18-s + (−0.5 − 0.866i)19-s + 0.999i·22-s + (−0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + (−0.866 − 0.499i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7301272550\)
\(L(\frac12)\) \(\approx\) \(0.7301272550\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.759173444433636412719518089686, −8.733961700566874428469052993831, −8.022743633158296332981078315651, −7.40777147338803644368042681711, −6.62183338378830887072184456514, −5.70991421540247825909762392288, −4.79624190615770395210652735169, −3.65014182070340203261061302255, −2.22789076054254731537704122915, −0.859251812738902875620641057435, 1.56204309910663500106967594800, 2.41174021469771824549129128864, 3.74847151544128625713042617669, 4.46019147862406910841822449111, 6.04765220415516302250772712997, 6.65530479268562245698338828513, 7.50153279149318724578267465616, 8.550766275504815239030674724939, 9.052848179829081174135367203866, 9.821713823392599098179848893107

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