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Properties

Label 2-440-440.339-c0-0-0
Degree $2$
Conductor $440$
Sign $0.624 + 0.781i$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (1.30 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 0.999·10-s + (−0.809 − 0.587i)11-s + (0.190 − 0.587i)13-s + (1.30 − 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.5 + 0.363i)19-s + (0.309 − 0.951i)20-s + (−0.809 + 0.587i)22-s − 1.61·23-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (1.30 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + 0.999·10-s + (−0.809 − 0.587i)11-s + (0.190 − 0.587i)13-s + (1.30 − 0.951i)14-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)18-s + (−0.5 + 0.363i)19-s + (0.309 − 0.951i)20-s + (−0.809 + 0.587i)22-s − 1.61·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(0.219588\)
Root analytic conductor: \(0.468602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :0),\ 0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.012779550\)
\(L(\frac12)\) \(\approx\) \(1.012779550\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−11.12843514564958988751007606460, −10.58974354698900865759302842388, −9.654357997484594245754386976708, −8.622618552708444610600095112258, −7.78146330002105350293180902704, −6.07227384501106205875941073052, −5.56492710492019753904178142926, −4.16355447724190280730310117905, −2.95940450687167707467735651702, −1.90361583526390890951689268277, 1.88027193216260198049203875648, 4.30406733288640467429795235764, 4.65731196269482711007823960314, 5.59965569299876557005614545241, 6.96644416191614110142747530473, 8.035039239007306342141434803689, 8.183983189476524650027166011426, 9.574637777965322706296818329343, 10.45462682551343250004835080605, 11.55952278169472202022559021367

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