Properties

Label 2-1400-1.1-c1-0-1
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 + 2·17-s − 2·19-s + 2·21-s − 8·23-s + 4·27-s + 2·29-s + 4·31-s + 6·37-s − 2·41-s − 8·43-s + 4·47-s + 49-s − 4·51-s + 10·53-s + 4·57-s + 6·59-s + 4·61-s − 63-s + 12·67-s + 16·69-s + 14·73-s − 8·79-s − 11·81-s − 6·83-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.485·17-s − 0.458·19-s + 0.436·21-s − 1.66·23-s + 0.769·27-s + 0.371·29-s + 0.718·31-s + 0.986·37-s − 0.312·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.560·51-s + 1.37·53-s + 0.529·57-s + 0.781·59-s + 0.512·61-s − 0.125·63-s + 1.46·67-s + 1.92·69-s + 1.63·73-s − 0.900·79-s − 1.22·81-s − 0.658·83-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

\(L(1)\) \(\approx\) \(0.8479340745\)
\(L(\frac12)\) \(\approx\) \(0.8479340745\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.962370660926456280013102621248, −8.694697440739683427951548718991, −7.965971686397333220186526353962, −6.86591116390976664813936408049, −6.19535765189485358396007496237, −5.55399016745426097333734158356, −4.61889361197427675528739680931, −3.62433266137155427382097354955, −2.29482126376589182159266782001, −0.68915328699946070952810996679, 0.68915328699946070952810996679, 2.29482126376589182159266782001, 3.62433266137155427382097354955, 4.61889361197427675528739680931, 5.55399016745426097333734158356, 6.19535765189485358396007496237, 6.86591116390976664813936408049, 7.965971686397333220186526353962, 8.694697440739683427951548718991, 9.962370660926456280013102621248

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