Properties

Label 2-1400-1.1-c1-0-4
Degree $2$
Conductor $1400$
Sign $1$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.37·3-s + 7-s + 2.62·9-s + 6.37·11-s − 4.37·13-s + 0.372·17-s − 4.74·19-s − 2.37·21-s + 4.74·23-s + 0.883·27-s − 4.37·29-s − 8·31-s − 15.1·33-s + 2·37-s + 10.3·39-s + 6.74·41-s + 8.74·43-s + 7.11·47-s + 49-s − 0.883·51-s − 10.7·53-s + 11.2·57-s + 8·59-s − 2.74·61-s + 2.62·63-s + 4·67-s − 11.2·69-s + ⋯
L(s)  = 1  − 1.36·3-s + 0.377·7-s + 0.875·9-s + 1.92·11-s − 1.21·13-s + 0.0902·17-s − 1.08·19-s − 0.517·21-s + 0.989·23-s + 0.169·27-s − 0.811·29-s − 1.43·31-s − 2.63·33-s + 0.328·37-s + 1.66·39-s + 1.05·41-s + 1.33·43-s + 1.03·47-s + 0.142·49-s − 0.123·51-s − 1.47·53-s + 1.49·57-s + 1.04·59-s − 0.351·61-s + 0.331·63-s + 0.488·67-s − 1.35·69-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: no
Arithmetic: yes
Character: $\chi_{1400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.036787562\)
\(L(\frac12)\) \(\approx\) \(1.036787562\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.37T + 3T^{2} \)
11 \( 1 - 6.37T + 11T^{2} \)
13 \( 1 + 4.37T + 13T^{2} \)
17 \( 1 - 0.372T + 17T^{2} \)
19 \( 1 + 4.74T + 19T^{2} \)
23 \( 1 - 4.74T + 23T^{2} \)
29 \( 1 + 4.37T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 6.74T + 41T^{2} \)
43 \( 1 - 8.74T + 43T^{2} \)
47 \( 1 - 7.11T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 2.74T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 9.48T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 9.86T + 97T^{2} \)
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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

−9.397991707027163850618833447413, −9.109179329456040175650741661490, −7.72950235521558262835445175074, −6.91045554305168647134119697223, −6.28683622552320494088663864291, −5.43099998452876702819854300790, −4.61292101077022212155390767498, −3.79847429204527997960070107939, −2.11361203192552443151639818170, −0.800014161814407391550046830876, 0.800014161814407391550046830876, 2.11361203192552443151639818170, 3.79847429204527997960070107939, 4.61292101077022212155390767498, 5.43099998452876702819854300790, 6.28683622552320494088663864291, 6.91045554305168647134119697223, 7.72950235521558262835445175074, 9.109179329456040175650741661490, 9.397991707027163850618833447413

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