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Properties

Label 2-2800-7.6-c0-0-1
Degree $2$
Conductor $2800$
Sign $1$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 9-s + 11-s + 23-s − 29-s − 37-s + 43-s + 49-s + 2·53-s − 63-s + 67-s + 71-s − 77-s + 79-s + 81-s + 99-s − 2·107-s − 109-s − 113-s + ⋯
L(s)  = 1  − 7-s + 9-s + 11-s + 23-s − 29-s − 37-s + 43-s + 49-s + 2·53-s − 63-s + 67-s + 71-s − 77-s + 79-s + 81-s + 99-s − 2·107-s − 109-s − 113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2800} (2001, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267412057\)
\(L(\frac12)\) \(\approx\) \(1.267412057\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
11 \( 1 - T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 - T + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 - T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.302520512005021771005543871914, −8.293002200417010951718373170258, −7.09639356686388710507001903671, −6.96380723522820303532530068493, −6.01585134906469945856124381971, −5.12648868804480721874910205154, −4.02941000849521646912602619330, −3.55907555713187704293632323490, −2.33393186025838007092970621980, −1.10130977861732563472765543185, 1.10130977861732563472765543185, 2.33393186025838007092970621980, 3.55907555713187704293632323490, 4.02941000849521646912602619330, 5.12648868804480721874910205154, 6.01585134906469945856124381971, 6.96380723522820303532530068493, 7.09639356686388710507001903671, 8.293002200417010951718373170258, 9.302520512005021771005543871914

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