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Properties

Label 2-2240-280.27-c0-0-2
Degree $2$
Conductor $2240$
Sign $0.811 + 0.584i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 1.30i)3-s + (0.923 − 0.382i)5-s + (−0.707 + 0.707i)7-s + 2.41i·9-s + (−0.541 + 0.541i)13-s + (−1.70 − 0.707i)15-s + 1.84·19-s + 1.84·21-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (1.84 − 1.84i)27-s + (−0.382 + 0.923i)35-s + 1.41·39-s + (0.923 + 2.23i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (−1.30 − 1.30i)3-s + (0.923 − 0.382i)5-s + (−0.707 + 0.707i)7-s + 2.41i·9-s + (−0.541 + 0.541i)13-s + (−1.70 − 0.707i)15-s + 1.84·19-s + 1.84·21-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (1.84 − 1.84i)27-s + (−0.382 + 0.923i)35-s + 1.41·39-s + (0.923 + 2.23i)45-s − 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ 0.811 + 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8282947946\)
\(L(\frac12)\) \(\approx\) \(0.8282947946\)
\(L(1)\) 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 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - 1.84T + T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 0.765T + T^{2} \)
61 \( 1 - 0.765T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.363856363739438027564031475265, −8.262098627011292838066640537586, −7.20074620214003104610056502331, −6.86131577518638240665159867280, −5.87595821499349437977809058814, −5.49998677075754978004170270929, −4.83490276151516256354570777762, −3.03144636401476556685171840867, −2.01160902141762056351134074862, −1.06358220914509334243473312442, 0.876738562723511710318492258497, 2.89966319536100283246237814666, 3.59720474501798508937806898044, 4.76125513273829539354584832256, 5.26330357856704972176084657795, 6.06013419176642202103154997750, 6.71606494708682384281027721915, 7.47510084968351190136998827597, 8.999385275018855700286166246172, 9.665030040825505519828374728796

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