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Properties

Label 2-40-8.5-c3-0-6
Degree $2$
Conductor $40$
Sign $0.674 - 0.738i$
Analytic cond. $2.36007$
Root an. cond. $1.53625$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.74 + 0.690i)2-s + 4.24i·3-s + (7.04 + 3.78i)4-s − 5i·5-s + (−2.93 + 11.6i)6-s − 14.6·7-s + (16.7 + 15.2i)8-s + 8.98·9-s + (3.45 − 13.7i)10-s − 14.9i·11-s + (−16.0 + 29.9i)12-s − 85.6i·13-s + (−40.1 − 10.1i)14-s + 21.2·15-s + (35.2 + 53.4i)16-s − 91.9·17-s + ⋯
L(s)  = 1  + (0.969 + 0.244i)2-s + 0.816i·3-s + (0.880 + 0.473i)4-s − 0.447i·5-s + (−0.199 + 0.792i)6-s − 0.789·7-s + (0.738 + 0.674i)8-s + 0.332·9-s + (0.109 − 0.433i)10-s − 0.411i·11-s + (−0.386 + 0.719i)12-s − 1.82i·13-s + (−0.766 − 0.192i)14-s + 0.365·15-s + (0.551 + 0.834i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.674 - 0.738i$
Analytic conductor: \(2.36007\)
Root analytic conductor: \(1.53625\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :3/2),\ 0.674 - 0.738i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.90820 + 0.841348i\)
\(L(\frac12)\) \(\approx\) \(1.90820 + 0.841348i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.74 - 0.690i)T \)
5 \( 1 + 5iT \)
good3 \( 1 - 4.24iT - 27T^{2} \)
7 \( 1 + 14.6T + 343T^{2} \)
11 \( 1 + 14.9iT - 1.33e3T^{2} \)
13 \( 1 + 85.6iT - 2.19e3T^{2} \)
17 \( 1 + 91.9T + 4.91e3T^{2} \)
19 \( 1 - 60.3iT - 6.85e3T^{2} \)
23 \( 1 - 1.33T + 1.21e4T^{2} \)
29 \( 1 + 25.5iT - 2.43e4T^{2} \)
31 \( 1 - 73.4T + 2.97e4T^{2} \)
37 \( 1 - 211. iT - 5.06e4T^{2} \)
41 \( 1 - 330.T + 6.89e4T^{2} \)
43 \( 1 - 388. iT - 7.95e4T^{2} \)
47 \( 1 + 550.T + 1.03e5T^{2} \)
53 \( 1 + 187. iT - 1.48e5T^{2} \)
59 \( 1 + 779. iT - 2.05e5T^{2} \)
61 \( 1 - 358. iT - 2.26e5T^{2} \)
67 \( 1 + 283. iT - 3.00e5T^{2} \)
71 \( 1 + 534.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3T + 3.89e5T^{2} \)
79 \( 1 - 1.11e3T + 4.93e5T^{2} \)
83 \( 1 + 1.19e3iT - 5.71e5T^{2} \)
89 \( 1 + 398.T + 7.04e5T^{2} \)
97 \( 1 - 278.T + 9.12e5T^{2} \)
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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

−15.76502144841173281244013663231, −14.90910569666006557752599321571, −13.28154713153680944510404641422, −12.67389265342340356576528912419, −11.02490631259592728494965818678, −9.852847255390021581308170729796, −8.082376707219367425694489157275, −6.26194827591781492820306057514, −4.82363312095553163575400885407, −3.33484907824927590865671573557, 2.18929299313766620482508560583, 4.27822092329348926575179365061, 6.51665500573088359576443933196, 7.05594901291645519339273332756, 9.481609636161980196123799527119, 11.04155167966158107180762221049, 12.18278593445338284060310595189, 13.20359005808868035515698449924, 14.00102048443758378897872593763, 15.35641908478215283833587704174

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