Properties

Label 2-2240-56.27-c1-0-4
Degree $2$
Conductor $2240$
Sign $-0.0108 + 0.999i$
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  + 2.84i·3-s − 5-s + (−0.712 + 2.54i)7-s − 5.08·9-s − 5.41·11-s + 0.641·13-s − 2.84i·15-s + 5.16i·17-s + 3.44i·19-s + (−7.24 − 2.02i)21-s + 7.94i·23-s + 25-s − 5.94i·27-s − 10.0i·29-s + 5.38·31-s + ⋯
L(s)  = 1  + 1.64i·3-s − 0.447·5-s + (−0.269 + 0.963i)7-s − 1.69·9-s − 1.63·11-s + 0.177·13-s − 0.734i·15-s + 1.25i·17-s + 0.790i·19-s + (−1.58 − 0.442i)21-s + 1.65i·23-s + 0.200·25-s − 1.14i·27-s − 1.86i·29-s + 0.966·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0108 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0108 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.0108 + 0.999i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.0108 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5072608845\)
\(L(\frac12)\) \(\approx\) \(0.5072608845\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (0.712 - 2.54i)T \)
good3 \( 1 - 2.84iT - 3T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 - 0.641T + 13T^{2} \)
17 \( 1 - 5.16iT - 17T^{2} \)
19 \( 1 - 3.44iT - 19T^{2} \)
23 \( 1 - 7.94iT - 23T^{2} \)
29 \( 1 + 10.0iT - 29T^{2} \)
31 \( 1 - 5.38T + 31T^{2} \)
37 \( 1 - 0.931iT - 37T^{2} \)
41 \( 1 + 8.33iT - 41T^{2} \)
43 \( 1 + 6.76T + 43T^{2} \)
47 \( 1 - 8.86T + 47T^{2} \)
53 \( 1 - 6.31iT - 53T^{2} \)
59 \( 1 + 11.5iT - 59T^{2} \)
61 \( 1 - 0.834T + 61T^{2} \)
67 \( 1 + 2.01T + 67T^{2} \)
71 \( 1 - 0.628iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + 2.98iT - 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + 13.3iT - 89T^{2} \)
97 \( 1 - 9.07iT - 97T^{2} \)
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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

−9.862340052199966568194670993734, −8.853084505438688010700870720338, −8.257333406950046958174494659076, −7.61038257771558316259316318918, −6.02959231609745097397731336280, −5.62192143809436693277665641451, −4.81905214375518929022173785706, −3.89703323671146191454950755804, −3.24428630734616561976454734882, −2.22457243425545873051792133504, 0.19786460350329547695592931922, 1.04406486672315284661363169514, 2.53675800359527101060742319959, 3.04344404820361631122910962950, 4.55656258551274126837968026400, 5.28963916577635910804629127438, 6.54246438293579499179795061471, 6.94260181684950600562795945277, 7.60348970237964786344777645594, 8.172302147108561998009396223801

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