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Properties

Label 2-440-440.179-c0-0-1
Degree $2$
Conductor $440$
Sign $0.794 + 0.606i$
Analytic cond. $0.219588$
Root an. cond. $0.468602$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.190 + 0.587i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s + (0.309 − 0.951i)11-s + (−1.30 + 0.951i)13-s + (0.190 + 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 + 0.951i)18-s + (−0.5 − 1.53i)19-s + (0.809 − 0.587i)20-s + (−0.309 − 0.951i)22-s − 0.618·23-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.190 + 0.587i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s + (0.309 − 0.951i)11-s + (−1.30 + 0.951i)13-s + (0.190 + 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 + 0.951i)18-s + (−0.5 − 1.53i)19-s + (0.809 − 0.587i)20-s + (−0.309 − 0.951i)22-s − 0.618·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(0.219588\)
Root analytic conductor: \(0.468602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :0),\ 0.794 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.300079757\)
\(L(\frac12)\) \(\approx\) \(1.300079757\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−11.25475604055869658721359625719, −10.67139302993169669683955771630, −9.487982224496555184867322442778, −8.958007632825329462866202510035, −7.25973080931310370341923324072, −6.22346206455128395506322609408, −5.57410129845919380861143788157, −4.46077191328262633763758848575, −2.83680942965247761015056554477, −2.27564294856788980626939293295, 2.27731767776347507384128904009, 3.69560336399986196602372267883, 4.83982128833857048520652010632, 5.72702167470718634935834332459, 6.57740941660466985650204368214, 7.65393167476704904242110573217, 8.529247801608341255183869138501, 9.706022418258114375571957588571, 10.36808009978270632317216429952, 11.99127969564196527604844114156

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