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Properties

Label 2-1440-20.7-c1-0-14
Degree $2$
Conductor $1440$
Sign $0.553 + 0.832i$
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  + (−2.23 + 0.0743i)5-s + (−3.16 + 3.16i)7-s − 4.46i·11-s + (−2.51 + 2.51i)13-s + (2.30 + 2.30i)17-s + 2.61·19-s + (−1.64 − 1.64i)23-s + (4.98 − 0.332i)25-s − 8.17i·29-s − 4i·31-s + (6.82 − 7.29i)35-s + (5.80 + 5.80i)37-s + 2.61·41-s + (−5.14 − 5.14i)43-s + (−0.679 + 0.679i)47-s + ⋯
L(s)  = 1  + (−0.999 + 0.0332i)5-s + (−1.19 + 1.19i)7-s − 1.34i·11-s + (−0.698 + 0.698i)13-s + (0.560 + 0.560i)17-s + 0.600·19-s + (−0.342 − 0.342i)23-s + (0.997 − 0.0664i)25-s − 1.51i·29-s − 0.718i·31-s + (1.15 − 1.23i)35-s + (0.953 + 0.953i)37-s + 0.408·41-s + (−0.785 − 0.785i)43-s + (−0.0991 + 0.0991i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7875334508\)
\(L(\frac12)\) \(\approx\) \(0.7875334508\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0743i)T \)
good7 \( 1 + (3.16 - 3.16i)T - 7iT^{2} \)
11 \( 1 + 4.46iT - 11T^{2} \)
13 \( 1 + (2.51 - 2.51i)T - 13iT^{2} \)
17 \( 1 + (-2.30 - 2.30i)T + 17iT^{2} \)
19 \( 1 - 2.61T + 19T^{2} \)
23 \( 1 + (1.64 + 1.64i)T + 23iT^{2} \)
29 \( 1 + 8.17iT - 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (-5.80 - 5.80i)T + 37iT^{2} \)
41 \( 1 - 2.61T + 41T^{2} \)
43 \( 1 + (5.14 + 5.14i)T + 43iT^{2} \)
47 \( 1 + (0.679 - 0.679i)T - 47iT^{2} \)
53 \( 1 + (-7.81 + 7.81i)T - 53iT^{2} \)
59 \( 1 + 4.88T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + (-9.44 + 9.44i)T - 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (5.61 - 5.61i)T - 73iT^{2} \)
79 \( 1 + 3.57T + 79T^{2} \)
83 \( 1 + (1.34 + 1.34i)T + 83iT^{2} \)
89 \( 1 - 17.6iT - 89T^{2} \)
97 \( 1 + (1.32 + 1.32i)T + 97iT^{2} \)
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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

−9.458763128482220012492765596306, −8.468056623363762033901222354808, −8.031924410198691698978099979529, −6.86335556842210820371871187742, −6.13462327527197140547542091126, −5.39519596355531241577511355868, −4.10600276774436803416387616467, −3.29343244981624514703763985793, −2.44758628950062581630952013505, −0.41059617860480259465757691379, 0.949331926277693430179547742567, 2.83626204945136180717549505304, 3.62738999935718580262673084787, 4.47586898626887715671959160414, 5.37149131681298648721537746789, 6.75213668202640915567474380299, 7.35555684946793651116562542046, 7.64489004141055435935909053484, 8.946754082329899326234742045487, 9.919617945806685512232829534176

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