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Properties

Label 2-2800-5.4-c1-0-6
Degree $2$
Conductor $2800$
Sign $-0.894 + 0.447i$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + i·7-s − 6·9-s + 5·11-s − 5i·13-s + 7i·17-s − 2·19-s − 3·21-s + 2i·23-s − 9i·27-s − 7·29-s − 4·31-s + 15i·33-s + 6i·37-s + 15·39-s + ⋯
L(s)  = 1  + 1.73i·3-s + 0.377i·7-s − 2·9-s + 1.50·11-s − 1.38i·13-s + 1.69i·17-s − 0.458·19-s − 0.654·21-s + 0.417i·23-s − 1.73i·27-s − 1.29·29-s − 0.718·31-s + 2.61i·33-s + 0.986i·37-s + 2.40·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $-0.894 + 0.447i$
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: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093434277\)
\(L(\frac12)\) \(\approx\) \(1.093434277\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT - 3T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 13iT - 97T^{2} \)
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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

−9.331243751693377777695890021028, −8.588781150376436725579490489294, −8.139583181279148725646084066985, −6.79461828124479501283005807905, −5.84709066790486068284477349870, −5.42361587279815601354953828351, −4.36241471687044704903841453917, −3.72571509984016031491542950728, −3.15789904145046700917368455608, −1.65617571331981277750848083600, 0.34047552422091485053748234609, 1.54626619684282501890904015405, 2.13466988899724902419132066078, 3.41272955473304130281038843477, 4.36903042101480602545038260618, 5.48625451372435419391067603093, 6.46918025374441034590737957831, 6.92881110672467810626490745442, 7.28323647905176075416117736055, 8.260569503153250233009830913617

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