Properties

Label 2-1400-1.1-c1-0-7
Degree $2$
Conductor $1400$
Sign $1$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 3·9-s − 4·11-s − 2·13-s + 6·17-s + 8·19-s + 6·29-s + 8·31-s + 2·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s − 6·53-s − 6·61-s − 3·63-s + 4·67-s − 8·71-s − 10·73-s − 4·77-s + 16·79-s + 9·81-s − 8·83-s − 6·89-s − 2·91-s + 6·97-s + 12·99-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s − 1.20·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.768·61-s − 0.377·63-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s + 1.80·79-s + 81-s − 0.878·83-s − 0.635·89-s − 0.209·91-s + 0.609·97-s + 1.20·99-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.564439584\)
\(L(\frac12)\) \(\approx\) \(1.564439584\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−9.694440338392361408863185872440, −8.660773281175179322159496718137, −7.81362030580797675551267597140, −7.47489161542833731239859573103, −6.06292563014554710285489032478, −5.38565578310165680979646788554, −4.71019542661529810418780789156, −3.18975677352219866037498641634, −2.62755636242588070812542768623, −0.916171764469637391994006820348, 0.916171764469637391994006820348, 2.62755636242588070812542768623, 3.18975677352219866037498641634, 4.71019542661529810418780789156, 5.38565578310165680979646788554, 6.06292563014554710285489032478, 7.47489161542833731239859573103, 7.81362030580797675551267597140, 8.660773281175179322159496718137, 9.694440338392361408863185872440

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