Properties

Label 2-1440-45.14-c0-0-1
Degree $2$
Conductor $1440$
Sign $-0.642 - 0.766i$
Analytic cond. $0.718653$
Root an. cond. $0.847734$
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  + (0.258 + 0.965i)3-s + (0.866 + 0.5i)5-s + (−1.67 + 0.965i)7-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)15-s + (−1.36 − 1.36i)21-s + (0.258 − 0.448i)23-s + (0.499 + 0.866i)25-s + (−0.707 − 0.707i)27-s + (−1.5 + 0.866i)29-s − 1.93·35-s + (0.866 + 0.5i)41-s + (1.22 − 0.707i)43-s − 45-s + (0.965 + 1.67i)47-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)3-s + (0.866 + 0.5i)5-s + (−1.67 + 0.965i)7-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)15-s + (−1.36 − 1.36i)21-s + (0.258 − 0.448i)23-s + (0.499 + 0.866i)25-s + (−0.707 − 0.707i)27-s + (−1.5 + 0.866i)29-s − 1.93·35-s + (0.866 + 0.5i)41-s + (1.22 − 0.707i)43-s − 45-s + (0.965 + 1.67i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(0.718653\)
Root analytic conductor: \(0.847734\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :0),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.009998171\)
\(L(\frac12)\) \(\approx\) \(1.009998171\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
good7 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - 1.73iT - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.835588459437010038800086617425, −9.193941114236131850617740382140, −8.999821434996974316059061531390, −7.58626575487523376451939704978, −6.49215325626714137311195338876, −5.89743915031330638244984415874, −5.20558638680078100139719807957, −3.84233585379498582247354340656, −2.99682199180946684505409853231, −2.33823166073465499462402561241, 0.77434470585115840678542513468, 2.15355109339415842354511625985, 3.19814834111105719778202090707, 4.16371728020370826332622017686, 5.72332067329311014745171409718, 6.10934655610387730494938468986, 7.09456604592271074832058776497, 7.55141499164128436458963486955, 8.828330402671414766740953393872, 9.367432426058290177820669217320

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