Properties

Label 2-1400-56.13-c0-0-8
Degree $2$
Conductor $1400$
Sign $-1$
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 − 7-s − 0.999·8-s − 9-s − 1.73i·11-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−1.49 − 0.866i)22-s − 23-s + (0.499 + 0.866i)28-s − 1.73i·29-s + (0.499 + 0.866i)32-s + (0.499 + 0.866i)36-s + 1.73i·37-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 7-s − 0.999·8-s − 9-s − 1.73i·11-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−1.49 − 0.866i)22-s − 23-s + (0.499 + 0.866i)28-s − 1.73i·29-s + (0.499 + 0.866i)32-s + (0.499 + 0.866i)36-s + 1.73i·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7493030579\)
\(L(\frac12)\) \(\approx\) \(0.7493030579\)
\(L(1)\) 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 \( 1 + T \)
good3 \( 1 + T^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
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

−9.493651268341040971641532956191, −8.704452215552103367009069110378, −8.044919118323763129273374989878, −6.37055667514839173299920024060, −6.06148820894904689528403539727, −5.20451022881704267442759786319, −3.85084614058781189252424500176, −3.24363039444552861579197778460, −2.36349393864433788223460608726, −0.49418645872110009508556534167, 2.37007642889111239523378475339, 3.40098446496291043402696903894, 4.33499180401528034990351055655, 5.29672123352404281747730163147, 6.07680643003824312163113423068, 6.89467839796931541184747272690, 7.50665489231399357733332891726, 8.469937811031851222126383239222, 9.303160052920402604799326578851, 9.843954593408458694068303863593

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