Properties

Label 2-2800-5.4-c1-0-37
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + i·7-s + 2·9-s − 3·11-s + 2i·13-s − 3i·17-s − 7·19-s + 21-s − 5i·27-s + 6·29-s + 4·31-s + 3i·33-s − 8i·37-s + 2·39-s − 9·41-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.377i·7-s + 0.666·9-s − 0.904·11-s + 0.554i·13-s − 0.727i·17-s − 1.60·19-s + 0.218·21-s − 0.962i·27-s + 1.11·29-s + 0.718·31-s + 0.522i·33-s − 1.31i·37-s + 0.320·39-s − 1.40·41-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: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.201589651\)
\(L(\frac12)\) \(\approx\) \(1.201589651\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 3T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 5iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.439535071527301795498875791572, −7.86524564748114601378557105422, −6.83119856000446074639433272428, −6.59758426456412457724930490957, −5.42292008102708855710365113617, −4.70439236326913879374483276491, −3.78897014058030505783461895990, −2.50515243931651445378355835746, −1.90377924917026578341818335447, −0.39297841375596980079219769807, 1.26353506707511164888895936092, 2.55441391864240419216153383205, 3.49676909267743451768052399893, 4.50931814997606011834262642095, 4.84924149369112346935189782289, 6.08198688056887357232281118306, 6.65520217491701340549277577520, 7.71813709941001194391347837439, 8.234510508171584789089015834907, 9.017369554046076493311831701931

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