Properties

Label 2-1440-12.11-c1-0-0
Degree $2$
Conductor $1440$
Sign $-0.985 + 0.169i$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s + 3.41i·7-s − 2.58·11-s − 3.41·13-s − 1.17i·17-s − 4.82·23-s − 25-s − 6i·29-s + 6.48i·31-s − 3.41·35-s − 9.07·37-s − 11.0i·41-s − 6.82i·43-s + 5.65·47-s − 4.65·49-s + ⋯
L(s)  = 1  + 0.447i·5-s + 1.29i·7-s − 0.779·11-s − 0.946·13-s − 0.284i·17-s − 1.00·23-s − 0.200·25-s − 1.11i·29-s + 1.16i·31-s − 0.577·35-s − 1.49·37-s − 1.72i·41-s − 1.04i·43-s + 0.825·47-s − 0.665·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.985 + 0.169i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3379721590\)
\(L(\frac12)\) \(\approx\) \(0.3379721590\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - iT \)
good7 \( 1 - 3.41iT - 7T^{2} \)
11 \( 1 + 2.58T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 + 1.17iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 6.48iT - 31T^{2} \)
37 \( 1 + 9.07T + 37T^{2} \)
41 \( 1 + 11.0iT - 41T^{2} \)
43 \( 1 + 6.82iT - 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + 1.17iT - 53T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 14.4iT - 79T^{2} \)
83 \( 1 + 9.17T + 83T^{2} \)
89 \( 1 - 4.24iT - 89T^{2} \)
97 \( 1 - 2.48T + 97T^{2} \)
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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

−10.07059232661073446077601059323, −9.074238608412405889362099180842, −8.433382451584547715549959545911, −7.50986061553617405595987370587, −6.77213987707723257656505294708, −5.61786075098206753695476386655, −5.25294005742020447763546785498, −3.94321465814082114241693158567, −2.70807452517557273754609515357, −2.13403615606214718347597235670, 0.12641982256569041312251386921, 1.59552567414014799363337751330, 2.94443855828417120921757167260, 4.09645573458607914689342383210, 4.77872358348729169092522851647, 5.72482668957035523704622096188, 6.79794682135626334160360276781, 7.60277155322429224672156892573, 8.100586217692038058966235330728, 9.183930417254190500444321806524

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