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Properties

Label 2-2640-165.32-c0-0-1
Degree $2$
Conductor $2640$
Sign $0.850 - 0.525i$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
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  − i·3-s + i·5-s − 9-s + i·11-s + 15-s + (1 + i)23-s − 25-s + i·27-s + 33-s + (1 + i)37-s − i·45-s + (1 − i)47-s + i·49-s + (1 + i)53-s − 55-s + ⋯
L(s)  = 1  − i·3-s + i·5-s − 9-s + i·11-s + 15-s + (1 + i)23-s − 25-s + i·27-s + 33-s + (1 + i)37-s − i·45-s + (1 − i)47-s + i·49-s + (1 + i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2640\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2640} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2640,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100136809\)
\(L(\frac12)\) \(\approx\) \(1.100136809\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 - iT \)
11 \( 1 - iT \)
good7 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1 + i)T - iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 + 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1 - i)T + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−9.116860833309517002819615869757, −8.123210318768993620290819102310, −7.35390883067528941921533154780, −7.04628990568526417407888939247, −6.19277237014089302109336502278, −5.45395123377967342358625761567, −4.32652555083261691255498412564, −3.16521798672557096450017852461, −2.46070281147325476209224970614, −1.41377236716518528948429580663, 0.76471182389922984965754147007, 2.43305431048229169328673252695, 3.47220290360374581580936433713, 4.27815165490137974156367595733, 5.02249602661076573201021651885, 5.69219203745232120381272038802, 6.46075558632355314650224849124, 7.74275228026128726661793781882, 8.438715386973890340291950721302, 9.041426662672020409164101690264

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