Properties

Label 2-1400-1.1-c1-0-0
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  − 1.76·3-s − 7-s + 0.103·9-s − 0.626·11-s − 5.49·13-s − 0.896·17-s + 6.38·19-s + 1.76·21-s + 3.72·23-s + 5.10·27-s − 7.87·29-s + 7.52·31-s + 1.10·33-s − 6·37-s + 9.67·39-s + 7.72·41-s − 1.72·43-s − 5.87·47-s + 49-s + 1.57·51-s + 6.77·53-s − 11.2·57-s − 0.593·59-s + 7.13·61-s − 0.103·63-s + 5.79·67-s − 6.56·69-s + ⋯
L(s)  = 1  − 1.01·3-s − 0.377·7-s + 0.0343·9-s − 0.188·11-s − 1.52·13-s − 0.217·17-s + 1.46·19-s + 0.384·21-s + 0.777·23-s + 0.982·27-s − 1.46·29-s + 1.35·31-s + 0.192·33-s − 0.986·37-s + 1.54·39-s + 1.20·41-s − 0.263·43-s − 0.857·47-s + 0.142·49-s + 0.221·51-s + 0.930·53-s − 1.49·57-s − 0.0773·59-s + 0.913·61-s − 0.0129·63-s + 0.707·67-s − 0.790·69-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

\(L(1)\) \(\approx\) \(0.8214227965\)
\(L(\frac12)\) \(\approx\) \(0.8214227965\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.76T + 3T^{2} \)
11 \( 1 + 0.626T + 11T^{2} \)
13 \( 1 + 5.49T + 13T^{2} \)
17 \( 1 + 0.896T + 17T^{2} \)
19 \( 1 - 6.38T + 19T^{2} \)
23 \( 1 - 3.72T + 23T^{2} \)
29 \( 1 + 7.87T + 29T^{2} \)
31 \( 1 - 7.52T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 7.72T + 41T^{2} \)
43 \( 1 + 1.72T + 43T^{2} \)
47 \( 1 + 5.87T + 47T^{2} \)
53 \( 1 - 6.77T + 53T^{2} \)
59 \( 1 + 0.593T + 59T^{2} \)
61 \( 1 - 7.13T + 61T^{2} \)
67 \( 1 - 5.79T + 67T^{2} \)
71 \( 1 - 5.52T + 71T^{2} \)
73 \( 1 - 3.72T + 73T^{2} \)
79 \( 1 + 5.67T + 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−9.708516420451342416624553156024, −8.918052796131110514510744961704, −7.71845745435832364682029767468, −7.08444132262771253789531127816, −6.22969908981643172555909007877, −5.26522351164895423757386557354, −4.88323502689938343491147495418, −3.45591462771770814431260870441, −2.40096966435580942129278789581, −0.66632876187762090243030286651, 0.66632876187762090243030286651, 2.40096966435580942129278789581, 3.45591462771770814431260870441, 4.88323502689938343491147495418, 5.26522351164895423757386557354, 6.22969908981643172555909007877, 7.08444132262771253789531127816, 7.71845745435832364682029767468, 8.918052796131110514510744961704, 9.708516420451342416624553156024

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