Properties

Label 2-1400-1400.909-c0-0-0
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 + (−1.69 + 0.550i)3-s + (−0.309 − 0.951i)4-s + (0.987 − 0.156i)5-s + (0.550 − 1.69i)6-s + i·7-s + (0.951 + 0.309i)8-s + (1.76 − 1.27i)9-s + (−0.453 + 0.891i)10-s + (1.04 + 1.44i)12-s + (−0.533 − 0.734i)13-s + (−0.809 − 0.587i)14-s + (−1.58 + 0.809i)15-s + (−0.809 + 0.587i)16-s + 2.17i·18-s + (0.437 − 1.34i)19-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−1.69 + 0.550i)3-s + (−0.309 − 0.951i)4-s + (0.987 − 0.156i)5-s + (0.550 − 1.69i)6-s + i·7-s + (0.951 + 0.309i)8-s + (1.76 − 1.27i)9-s + (−0.453 + 0.891i)10-s + (1.04 + 1.44i)12-s + (−0.533 − 0.734i)13-s + (−0.809 − 0.587i)14-s + (−1.58 + 0.809i)15-s + (−0.809 + 0.587i)16-s + 2.17i·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

\(L(\frac{1}{2})\) \(\approx\) \(0.5334853246\)
\(L(\frac12)\) \(\approx\) \(0.5334853246\)
\(L(1)\) 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 \( 1 + (-0.987 + 0.156i)T \)
7 \( 1 - iT \)
good3 \( 1 + (1.69 - 0.550i)T + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.59 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (-1.87 - 0.610i)T + (0.809 + 0.587i)T^{2} \)
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

−9.753638495809056993474341709429, −9.266741246916665889442171379276, −8.434893908033552939634135280393, −7.02339583223518683761057891285, −6.49811165894837154075354769717, −5.68196746854986235748231253042, −5.15386567858242339158051738118, −4.66544671154155959903862461080, −2.53143505801144383627364182538, −0.848698346682820657225015781619, 1.13247035947316375386959047337, 1.89028893538651765078316343077, 3.53710045936482609835318625135, 4.72877220264555750777201854537, 5.46952796956540956969927034815, 6.52847945258033698813903166469, 7.15987714266698040921126136347, 7.81219922315076095106805883165, 9.282743335944497682058166671573, 9.928243282263006047497088992151

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