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Properties

Label 2-2420-220.59-c0-0-14
Degree $2$
Conductor $2420$
Sign $-0.801 + 0.598i$
Analytic cond. $1.20773$
Root an. cond. $1.09897$
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  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.104 − 0.994i)5-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.866 − 0.5i)10-s + (0.587 − 0.809i)13-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + (−0.913 + 0.406i)20-s + (−0.978 + 0.207i)25-s + (−0.309 − 0.951i)26-s + (−0.535 − 1.64i)29-s + i·32-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.104 − 0.994i)5-s + (−0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.866 − 0.5i)10-s + (0.587 − 0.809i)13-s + (−0.809 + 0.587i)16-s + (−0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + (−0.913 + 0.406i)20-s + (−0.978 + 0.207i)25-s + (−0.309 − 0.951i)26-s + (−0.535 − 1.64i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2420\)    =    \(2^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-0.801 + 0.598i$
Analytic conductor: \(1.20773\)
Root analytic conductor: \(1.09897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2420} (2259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2420,\ (\ :0),\ -0.801 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.509311967\)
\(L(\frac12)\) \(\approx\) \(1.509311967\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (0.104 + 0.994i)T \)
11 \( 1 \)
good3 \( 1 + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.535 + 1.64i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-1.64 + 0.535i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.535 - 1.64i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (1.01 - 1.40i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.90 - 0.618i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (-1.01 + 1.40i)T + (-0.309 - 0.951i)T^{2} \)
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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

−9.027078341187776863869134705514, −8.145789164910429589948928964865, −7.41167701478110914738509285888, −6.16413962896799012756358761772, −5.54835492181337972597421226076, −4.52541707857498075881438642857, −4.28069956128433351371260939254, −3.01121970904348353769728012121, −1.96626193972691274927910780021, −0.872971390990113599407966490477, 1.89435142462664199256272273851, 3.20941112930836806321638546293, 3.88146637017771465227507809878, 4.57149937655406698559852082232, 5.78998640134509760255301405587, 6.46347584701803618351600647958, 6.97550230469182849176176839962, 7.61243607163687495781216155721, 8.619325885077212165593613118202, 9.218199159967221599881198447442

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