Properties

Label 2-1400-1.1-c1-0-2
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  − 1.56·3-s + 7-s − 0.561·9-s − 6.12·11-s + 2·13-s + 1.56·17-s + 3.56·19-s − 1.56·21-s + 1.43·23-s + 5.56·27-s + 3.43·29-s − 9.12·31-s + 9.56·33-s + 8.80·37-s − 3.12·39-s − 2.43·41-s + 6.56·43-s − 8.24·47-s + 49-s − 2.43·51-s − 1.12·53-s − 5.56·57-s + 11.3·59-s + 11.1·61-s − 0.561·63-s + 7.87·67-s − 2.24·69-s + ⋯
L(s)  = 1  − 0.901·3-s + 0.377·7-s − 0.187·9-s − 1.84·11-s + 0.554·13-s + 0.378·17-s + 0.817·19-s − 0.340·21-s + 0.299·23-s + 1.07·27-s + 0.638·29-s − 1.63·31-s + 1.66·33-s + 1.44·37-s − 0.500·39-s − 0.380·41-s + 1.00·43-s − 1.20·47-s + 0.142·49-s − 0.341·51-s − 0.154·53-s − 0.736·57-s + 1.48·59-s + 1.42·61-s − 0.0707·63-s + 0.962·67-s − 0.270·69-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

\(L(1)\) \(\approx\) \(1.011834037\)
\(L(\frac12)\) \(\approx\) \(1.011834037\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 3T^{2} \)
11 \( 1 + 6.12T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 1.56T + 17T^{2} \)
19 \( 1 - 3.56T + 19T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 - 3.43T + 29T^{2} \)
31 \( 1 + 9.12T + 31T^{2} \)
37 \( 1 - 8.80T + 37T^{2} \)
41 \( 1 + 2.43T + 41T^{2} \)
43 \( 1 - 6.56T + 43T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 + 1.12T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 7.87T + 67T^{2} \)
71 \( 1 - 1.68T + 71T^{2} \)
73 \( 1 - 6.43T + 73T^{2} \)
79 \( 1 + 5.68T + 79T^{2} \)
83 \( 1 + 1.31T + 83T^{2} \)
89 \( 1 - 9.80T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−9.764886541843858359071372728501, −8.623794184886935393997146175628, −7.928524300526593613706540030669, −7.16615429425786812669629194646, −6.03755799412375172387116788170, −5.38190326402091152015642817625, −4.85377048861373154030166071134, −3.44642196103307571227694096033, −2.37833347873945477786681427203, −0.74789096578835579845080643150, 0.74789096578835579845080643150, 2.37833347873945477786681427203, 3.44642196103307571227694096033, 4.85377048861373154030166071134, 5.38190326402091152015642817625, 6.03755799412375172387116788170, 7.16615429425786812669629194646, 7.928524300526593613706540030669, 8.623794184886935393997146175628, 9.764886541843858359071372728501

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