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Properties

Label 2-40-40.27-c5-0-15
Degree $2$
Conductor $40$
Sign $0.890 - 0.455i$
Analytic cond. $6.41535$
Root an. cond. $2.53285$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.49 + 5.07i)2-s + (10.6 − 10.6i)3-s + (−19.5 − 25.3i)4-s + (−15.3 + 53.7i)5-s + (27.3 + 80.3i)6-s + (159. − 159. i)7-s + (177. − 35.6i)8-s + 18.1i·9-s + (−234. − 212. i)10-s + 516.·11-s + (−475. − 62.0i)12-s + (147. + 147. i)13-s + (412. + 1.21e3i)14-s + (407. + 732. i)15-s + (−262. + 989. i)16-s + (890. + 890. i)17-s + ⋯
L(s)  = 1  + (−0.441 + 0.897i)2-s + (0.680 − 0.680i)3-s + (−0.609 − 0.792i)4-s + (−0.274 + 0.961i)5-s + (0.309 + 0.910i)6-s + (1.23 − 1.23i)7-s + (0.980 − 0.196i)8-s + 0.0746i·9-s + (−0.741 − 0.671i)10-s + 1.28·11-s + (−0.953 − 0.124i)12-s + (0.242 + 0.242i)13-s + (0.561 + 1.65i)14-s + (0.467 + 0.840i)15-s + (−0.256 + 0.966i)16-s + (0.746 + 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ 0.890 - 0.455i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.64348 + 0.396428i\)
\(L(\frac12)\) \(\approx\) \(1.64348 + 0.396428i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.49 - 5.07i)T \)
5 \( 1 + (15.3 - 53.7i)T \)
good3 \( 1 + (-10.6 + 10.6i)T - 243iT^{2} \)
7 \( 1 + (-159. + 159. i)T - 1.68e4iT^{2} \)
11 \( 1 - 516.T + 1.61e5T^{2} \)
13 \( 1 + (-147. - 147. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-890. - 890. i)T + 1.41e6iT^{2} \)
19 \( 1 + 1.00e3iT - 2.47e6T^{2} \)
23 \( 1 + (1.84e3 + 1.84e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 1.84e3T + 2.05e7T^{2} \)
31 \( 1 + 4.97e3iT - 2.86e7T^{2} \)
37 \( 1 + (-2.81e3 + 2.81e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 1.24e4T + 1.15e8T^{2} \)
43 \( 1 + (1.18e4 - 1.18e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.19e4 - 1.19e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-1.07e4 - 1.07e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 8.27e3iT - 7.14e8T^{2} \)
61 \( 1 - 2.84e4iT - 8.44e8T^{2} \)
67 \( 1 + (-9.25e3 - 9.25e3i)T + 1.35e9iT^{2} \)
71 \( 1 + 1.85e4iT - 1.80e9T^{2} \)
73 \( 1 + (5.83e4 - 5.83e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 1.05e4T + 3.07e9T^{2} \)
83 \( 1 + (-1.26e4 + 1.26e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 6.59e4iT - 5.58e9T^{2} \)
97 \( 1 + (-1.45e4 - 1.45e4i)T + 8.58e9iT^{2} \)
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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

−14.76178309385694108265829827162, −14.36832253940665436886690192157, −13.52388140949286676856689864825, −11.36372918050802976971870556234, −10.21498428021169127213741162576, −8.385366202169315546313175150882, −7.56524430933935831081984717242, −6.57182514710529356535224295823, −4.21214346793921687774311396049, −1.44012860147340636181709208129, 1.52822282240251573150935988729, 3.60871383636915089599138551971, 5.04087143316476323710373499113, 8.213412795816319921562198192356, 8.839531007426196293921126614756, 9.822887045575980610463986121204, 11.79772855804483177802319196825, 12.04827067819246703402782154062, 13.92759864167726754440702533234, 14.98691916119326339286047785775

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