Properties

Label 2-1400-1.1-c1-0-25
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  + 3-s + 7-s − 2·9-s − 5·11-s − 13-s − 3·17-s − 6·19-s + 21-s + 6·23-s − 5·27-s − 9·29-s − 5·33-s − 6·37-s − 39-s + 8·41-s − 6·43-s − 3·47-s + 49-s − 3·51-s + 12·53-s − 6·57-s + 8·59-s − 4·61-s − 2·63-s + 4·67-s + 6·69-s + 8·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s − 2/3·9-s − 1.50·11-s − 0.277·13-s − 0.727·17-s − 1.37·19-s + 0.218·21-s + 1.25·23-s − 0.962·27-s − 1.67·29-s − 0.870·33-s − 0.986·37-s − 0.160·39-s + 1.24·41-s − 0.914·43-s − 0.437·47-s + 1/7·49-s − 0.420·51-s + 1.64·53-s − 0.794·57-s + 1.04·59-s − 0.512·61-s − 0.251·63-s + 0.488·67-s + 0.722·69-s + 0.949·71-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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−8.923791913883665586804923257643, −8.462219312931430130295138543437, −7.66737852056154649182867009604, −6.87549713237471057055540036166, −5.67002771047620563739767976211, −5.02967376320291988227199382319, −3.92102309478332339985108610603, −2.76252169464449428978311315426, −2.08175715199719083636321296262, 0, 2.08175715199719083636321296262, 2.76252169464449428978311315426, 3.92102309478332339985108610603, 5.02967376320291988227199382319, 5.67002771047620563739767976211, 6.87549713237471057055540036166, 7.66737852056154649182867009604, 8.462219312931430130295138543437, 8.923791913883665586804923257643

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