Properties

Label 2-2240-5.4-c1-0-4
Degree $2$
Conductor $2240$
Sign $0.807 - 0.590i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  − 3.12i·3-s + (−1.32 − 1.80i)5-s + i·7-s − 6.76·9-s − 2.48·11-s + 4.15i·13-s + (−5.64 + 4.12i)15-s − 5.76i·17-s − 1.60·19-s + 3.12·21-s + 7.28i·23-s + (−1.51 + 4.76i)25-s + 11.7i·27-s + 1.45·29-s − 2.24·31-s + ⋯
L(s)  = 1  − 1.80i·3-s + (−0.590 − 0.807i)5-s + 0.377i·7-s − 2.25·9-s − 0.749·11-s + 1.15i·13-s + (−1.45 + 1.06i)15-s − 1.39i·17-s − 0.369·19-s + 0.681·21-s + 1.51i·23-s + (−0.303 + 0.952i)25-s + 2.26i·27-s + 0.270·29-s − 0.404·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.807 - 0.590i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.807 - 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3279753685\)
\(L(\frac12)\) \(\approx\) \(0.3279753685\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 1.80i)T \)
7 \( 1 - iT \)
good3 \( 1 + 3.12iT - 3T^{2} \)
11 \( 1 + 2.48T + 11T^{2} \)
13 \( 1 - 4.15iT - 13T^{2} \)
17 \( 1 + 5.76iT - 17T^{2} \)
19 \( 1 + 1.60T + 19T^{2} \)
23 \( 1 - 7.28iT - 23T^{2} \)
29 \( 1 - 1.45T + 29T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 5.28iT - 43T^{2} \)
47 \( 1 + 3.45iT - 47T^{2} \)
53 \( 1 - 9.21iT - 53T^{2} \)
59 \( 1 + 5.92T + 59T^{2} \)
61 \( 1 + 5.35T + 61T^{2} \)
67 \( 1 + 7.52iT - 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 - 7.28iT - 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 2.73iT - 97T^{2} \)
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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

−9.067874294732085362845385729613, −8.097116736218823410479275305295, −7.50801132310055337453798675510, −7.08972625240239142253184506215, −6.01080203938411877728600457688, −5.36837312840749345534117531804, −4.36723065633882974752625138575, −3.00332364357698378627119875471, −2.08956047455687522791673897413, −1.11383796772185616752951366813, 0.12339308480590164766582464196, 2.60194272619800707753019422572, 3.29741240262584375005363382817, 4.10487345572472373686447857700, 4.69815035448418291999517402964, 5.71722021637013372551713533291, 6.40724163083226587627638702450, 7.68725955671088480477740942961, 8.259471302329889271609872358474, 8.957338954655511563705954915907

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