Properties

Label 2-1400-1.1-c1-0-12
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  + 2·3-s + 7-s + 9-s + 5·11-s + 8·17-s − 2·19-s + 2·21-s − 7·23-s − 4·27-s − 3·29-s + 4·31-s + 10·33-s − 37-s − 2·41-s + 3·43-s + 6·47-s + 49-s + 16·51-s + 10·53-s − 4·57-s − 4·59-s − 6·61-s + 63-s + 13·67-s − 14·69-s + 5·71-s + 6·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.94·17-s − 0.458·19-s + 0.436·21-s − 1.45·23-s − 0.769·27-s − 0.557·29-s + 0.718·31-s + 1.74·33-s − 0.164·37-s − 0.312·41-s + 0.457·43-s + 0.875·47-s + 1/7·49-s + 2.24·51-s + 1.37·53-s − 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.125·63-s + 1.58·67-s − 1.68·69-s + 0.593·71-s + 0.702·73-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

\(L(1)\) \(\approx\) \(2.814561857\)
\(L(\frac12)\) \(\approx\) \(2.814561857\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−9.513870706542333340648205265315, −8.684803347503568812792785869548, −8.069078370838473547557932517577, −7.39070327434938193188335055166, −6.30047245941881715667629546480, −5.45788503200219529211336521976, −4.06748454385499559567921683687, −3.59222171566378595092146482354, −2.39874083819611908400458517412, −1.31531657098182471156833900193, 1.31531657098182471156833900193, 2.39874083819611908400458517412, 3.59222171566378595092146482354, 4.06748454385499559567921683687, 5.45788503200219529211336521976, 6.30047245941881715667629546480, 7.39070327434938193188335055166, 8.069078370838473547557932517577, 8.684803347503568812792785869548, 9.513870706542333340648205265315

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