Properties

Label 2-2240-280.27-c0-0-6
Degree $2$
Conductor $2240$
Sign $0.584 - 0.811i$
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

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

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.584 - 0.811i$
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: \((2,\ 2240,\ (\ :0),\ 0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.130159354\)
\(L(\frac12)\) \(\approx\) \(2.130159354\)
\(L(1)\) 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

−9.360167470381046417864243237670, −8.491634378082461904876701962228, −8.314454477797238770667561986655, −7.17910524557817010046985322830, −6.13441464666611674436008689649, −4.94813011624446114414288209781, −4.46122015604205483880223942863, −3.82891785088532818853878384556, −2.46647599469830836879211899381, −1.93149227660603585043539232570, 1.57739388883241993525740637278, 2.22732277193062947190906378954, 2.82591772903394585270448774798, 4.01694798831336901339960667600, 5.39951443164235891282806665855, 6.15032153446660348223053598019, 6.87465362474936734473817390348, 7.68256762225788007207850042747, 8.328087942465828633868684732252, 8.873729463810123636130653040940

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