Properties

Label 2-2800-5.4-c1-0-40
Degree $2$
Conductor $2800$
Sign $-0.447 + 0.894i$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·3-s i·7-s − 9-s − 5·11-s + 8i·17-s − 2·19-s + 2·21-s − 7i·23-s + 4i·27-s + 3·29-s − 4·31-s − 10i·33-s i·37-s − 2·41-s + 3i·43-s + ⋯
L(s)  = 1  + 1.15i·3-s − 0.377i·7-s − 0.333·9-s − 1.50·11-s + 1.94i·17-s − 0.458·19-s + 0.436·21-s − 1.45i·23-s + 0.769i·27-s + 0.557·29-s − 0.718·31-s − 1.74i·33-s − 0.164i·37-s − 0.312·41-s + 0.457i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ -0.447 + 0.894i)\)

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 \( 1 + iT \)
good3 \( 1 - 2iT - 3T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 8iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 7iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 3iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−8.413086864234618494873052695891, −8.133685211579967905936760095884, −7.01884629898722068219912796360, −6.15274325143092457654797980340, −5.28379601360898431998292633966, −4.54782058936440789788789985950, −3.90668781513470109811689282590, −2.99469315841616912960814690385, −1.84205090641577382071622873419, 0, 1.33611695416700141873696129895, 2.45718304305413496758433000770, 3.01712644156276181559076837877, 4.49333666902057687959281748280, 5.34752915786799963321742546700, 5.96027751751010371658031391778, 7.08094170044652930332358686339, 7.41331755143394449844783082373, 8.064499817756993566215652899860

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