Properties

Label 2-2800-1.1-c1-0-29
Degree $2$
Conductor $2800$
Sign $-1$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.44·3-s − 7-s + 2.99·9-s + 4.89·11-s − 4.44·13-s − 2·17-s − 1.55·19-s + 2.44·21-s + 2.89·23-s + 6.89·29-s − 8.89·31-s − 11.9·33-s − 2·37-s + 10.8·39-s − 1.10·41-s − 0.898·43-s + 8.89·47-s + 49-s + 4.89·51-s + 10.8·53-s + 3.79·57-s + 1.55·59-s + 3.55·61-s − 2.99·63-s − 8·67-s − 7.10·69-s + 1.10·71-s + ⋯
L(s)  = 1  − 1.41·3-s − 0.377·7-s + 0.999·9-s + 1.47·11-s − 1.23·13-s − 0.485·17-s − 0.355·19-s + 0.534·21-s + 0.604·23-s + 1.28·29-s − 1.59·31-s − 2.08·33-s − 0.328·37-s + 1.74·39-s − 0.171·41-s − 0.137·43-s + 1.29·47-s + 0.142·49-s + 0.685·51-s + 1.49·53-s + 0.503·57-s + 0.201·59-s + 0.454·61-s − 0.377·63-s − 0.977·67-s − 0.854·69-s + 0.130·71-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ -1)\)

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 \( 1 + 2.44T + 3T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
23 \( 1 - 2.89T + 23T^{2} \)
29 \( 1 - 6.89T + 29T^{2} \)
31 \( 1 + 8.89T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 1.10T + 41T^{2} \)
43 \( 1 + 0.898T + 43T^{2} \)
47 \( 1 - 8.89T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 - 3.55T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 + 2.89T + 73T^{2} \)
79 \( 1 + 6.89T + 79T^{2} \)
83 \( 1 + 2.44T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−8.615231778091495348346849328066, −7.17163881570688473865439000383, −6.92810538526476435072768523651, −6.12007128634609921791500494127, −5.39706195677561815522561806423, −4.60093446893466198134098845149, −3.85199279054058084326272293738, −2.54313912566700600959037110610, −1.21552053391431321987771072747, 0, 1.21552053391431321987771072747, 2.54313912566700600959037110610, 3.85199279054058084326272293738, 4.60093446893466198134098845149, 5.39706195677561815522561806423, 6.12007128634609921791500494127, 6.92810538526476435072768523651, 7.17163881570688473865439000383, 8.615231778091495348346849328066

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