Properties

Label 2-1400-1.1-c1-0-18
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 $1$

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: \(1\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−9.081765532002136523974019193764, −8.643211456083727468505351617365, −7.17140806224938930353179347537, −6.50350248745145303182029444341, −6.11581570891865448091668483981, −4.87924420949910647863837290955, −4.26941098581734613671549427432, −2.99357703198152362408807196649, −1.48172745371809944994373337707, 0, 1.48172745371809944994373337707, 2.99357703198152362408807196649, 4.26941098581734613671549427432, 4.87924420949910647863837290955, 6.11581570891865448091668483981, 6.50350248745145303182029444341, 7.17140806224938930353179347537, 8.643211456083727468505351617365, 9.081765532002136523974019193764

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