Properties

Label 2-1400-1.1-c1-0-13
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  + 3·3-s − 7-s + 6·9-s − 5·11-s + 5·13-s + 7·17-s − 2·19-s − 3·21-s + 2·23-s + 9·27-s + 7·29-s + 4·31-s − 15·33-s + 6·37-s + 15·39-s − 12·41-s + 2·43-s − 47-s + 49-s + 21·51-s − 6·57-s − 4·59-s + 4·61-s − 6·63-s − 8·67-s + 6·69-s − 6·73-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.377·7-s + 2·9-s − 1.50·11-s + 1.38·13-s + 1.69·17-s − 0.458·19-s − 0.654·21-s + 0.417·23-s + 1.73·27-s + 1.29·29-s + 0.718·31-s − 2.61·33-s + 0.986·37-s + 2.40·39-s − 1.87·41-s + 0.304·43-s − 0.145·47-s + 1/7·49-s + 2.94·51-s − 0.794·57-s − 0.520·59-s + 0.512·61-s − 0.755·63-s − 0.977·67-s + 0.722·69-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

\(L(1)\) \(\approx\) \(3.092699168\)
\(L(\frac12)\) \(\approx\) \(3.092699168\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−9.537842583629422034623468739439, −8.420602845795135280432161008655, −8.266382984595851713181898981059, −7.44876934013193343016755224328, −6.42644794024262941692165364434, −5.33447088843829441408193180318, −4.16608961416523430846434477431, −3.17358585952437131662296928437, −2.74217695950564727273864077608, −1.33365263604587907198019899672, 1.33365263604587907198019899672, 2.74217695950564727273864077608, 3.17358585952437131662296928437, 4.16608961416523430846434477431, 5.33447088843829441408193180318, 6.42644794024262941692165364434, 7.44876934013193343016755224328, 8.266382984595851713181898981059, 8.420602845795135280432161008655, 9.537842583629422034623468739439

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