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Properties

Label 2-2240-28.27-c1-0-44
Degree $2$
Conductor $2240$
Sign $0.991 - 0.127i$
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  + 2.96·3-s − i·5-s + (0.337 + 2.62i)7-s + 5.77·9-s + 3.63i·11-s − 4.77i·13-s − 2.96i·15-s − 4.77i·17-s + 4.57·19-s + (1 + 7.77i)21-s + 5.24i·23-s − 25-s + 8.20·27-s + 4.77·29-s − 5.92·31-s + ⋯
L(s)  = 1  + 1.70·3-s − 0.447i·5-s + (0.127 + 0.991i)7-s + 1.92·9-s + 1.09i·11-s − 1.32i·13-s − 0.764i·15-s − 1.15i·17-s + 1.04·19-s + (0.218 + 1.69i)21-s + 1.09i·23-s − 0.200·25-s + 1.58·27-s + 0.886·29-s − 1.06·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.991 - 0.127i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.991 - 0.127i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.566179608\)
\(L(\frac12)\) \(\approx\) \(3.566179608\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (-0.337 - 2.62i)T \)
good3 \( 1 - 2.96T + 3T^{2} \)
11 \( 1 - 3.63iT - 11T^{2} \)
13 \( 1 + 4.77iT - 13T^{2} \)
17 \( 1 + 4.77iT - 17T^{2} \)
19 \( 1 - 4.57T + 19T^{2} \)
23 \( 1 - 5.24iT - 23T^{2} \)
29 \( 1 - 4.77T + 29T^{2} \)
31 \( 1 + 5.92T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 - 2.02iT - 43T^{2} \)
47 \( 1 - 1.61T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 7.27T + 59T^{2} \)
61 \( 1 - 3.54iT - 61T^{2} \)
67 \( 1 - 12.5iT - 67T^{2} \)
71 \( 1 - 7.27iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 6.85iT - 79T^{2} \)
83 \( 1 - 2.02T + 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + 16.7iT - 97T^{2} \)
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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

−9.064211246450043739083718188052, −8.336899573813865019902217110022, −7.58967929773691043034175542941, −7.23265971055555412050814267312, −5.69785276102720878132452443101, −5.03938128838382431038144185586, −4.02686063657255858911776179009, −2.95185288801818529671805859050, −2.49898535829044665100496940515, −1.31711531901246847914922061443, 1.22502882199348718460214811621, 2.31801041745415224385082726989, 3.27813980272593856752649689563, 3.88031906566746860943621229698, 4.62758646659486025297238372726, 6.19510276646763606424091408752, 6.82765285765951534441475457749, 7.78294160058677135018377973463, 8.110204443518884165532621353126, 9.045437097461348914161718855670

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