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Properties

Label 2-1440-36.11-c1-0-3
Degree $2$
Conductor $1440$
Sign $-0.999 - 0.00833i$
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  + (0.793 + 1.53i)3-s + (0.866 − 0.5i)5-s + (−2.37 − 1.36i)7-s + (−1.74 + 2.44i)9-s + (−2.13 + 3.69i)11-s + (−1.63 − 2.83i)13-s + (1.45 + 0.936i)15-s + 2.83i·17-s + 0.727i·19-s + (0.226 − 4.73i)21-s + (0.0358 + 0.0620i)23-s + (0.499 − 0.866i)25-s + (−5.14 − 0.742i)27-s + (−2.08 − 1.20i)29-s + (−9.00 + 5.20i)31-s + ⋯
L(s)  = 1  + (0.458 + 0.888i)3-s + (0.387 − 0.223i)5-s + (−0.896 − 0.517i)7-s + (−0.580 + 0.814i)9-s + (−0.642 + 1.11i)11-s + (−0.454 − 0.787i)13-s + (0.376 + 0.241i)15-s + 0.687i·17-s + 0.166i·19-s + (0.0494 − 1.03i)21-s + (0.00747 + 0.0129i)23-s + (0.0999 − 0.173i)25-s + (−0.989 − 0.142i)27-s + (−0.386 − 0.223i)29-s + (−1.61 + 0.934i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.999 - 0.00833i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.999 - 0.00833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6212429202\)
\(L(\frac12)\) \(\approx\) \(0.6212429202\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.793 - 1.53i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
good7 \( 1 + (2.37 + 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.13 - 3.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.63 + 2.83i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.83iT - 17T^{2} \)
19 \( 1 - 0.727iT - 19T^{2} \)
23 \( 1 + (-0.0358 - 0.0620i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.08 + 1.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (9.00 - 5.20i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 + (3.93 - 2.27i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.31 + 3.06i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.66 - 6.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.55iT - 53T^{2} \)
59 \( 1 + (-0.127 - 0.221i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.20 - 9.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.10 - 2.94i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 + (-6.15 - 3.55i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.46 - 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.2iT - 89T^{2} \)
97 \( 1 + (4.75 - 8.23i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.942499396114275407000306246814, −9.385539689317016393470478750620, −8.389458620741190951103066420081, −7.62087869138315107815257090397, −6.74381666227380058456713389123, −5.57079798148172699605030870828, −4.92830008433498462488188525724, −3.89685521461838928942961459176, −3.07538017526814571868704717243, −1.96018295869907849640766793043, 0.21058063454917602052161553866, 1.94463313525043403876881863784, 2.83073653038639342996429009706, 3.57420226945350460479963069984, 5.19268753884486138423948138760, 6.01378236836400191264393271593, 6.67410337153189523214935216446, 7.46604002464685353577614500970, 8.315870589931771713403304693154, 9.309398550674181721876363706162

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