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Properties

Label 2-40-5.4-c5-0-4
Degree $2$
Conductor $40$
Sign $0.907 + 0.419i$
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  + 4.69i·3-s + (23.4 − 50.7i)5-s − 10.2i·7-s + 220.·9-s + 596.·11-s − 420. i·13-s + (238. + 109. i)15-s − 974. i·17-s + 380.·19-s + 48.1·21-s + 3.54e3i·23-s + (−2.02e3 − 2.37e3i)25-s + 2.17e3i·27-s − 5.44e3·29-s − 3.62e3·31-s + ⋯
L(s)  = 1  + 0.301i·3-s + (0.419 − 0.907i)5-s − 0.0791i·7-s + 0.909·9-s + 1.48·11-s − 0.690i·13-s + (0.273 + 0.126i)15-s − 0.817i·17-s + 0.241·19-s + 0.0238·21-s + 1.39i·23-s + (−0.648 − 0.760i)25-s + 0.574i·27-s − 1.20·29-s − 0.677·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.907 + 0.419i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ 0.907 + 0.419i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.79843 - 0.394959i\)
\(L(\frac12)\) \(\approx\) \(1.79843 - 0.394959i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-23.4 + 50.7i)T \)
good3 \( 1 - 4.69iT - 243T^{2} \)
7 \( 1 + 10.2iT - 1.68e4T^{2} \)
11 \( 1 - 596.T + 1.61e5T^{2} \)
13 \( 1 + 420. iT - 3.71e5T^{2} \)
17 \( 1 + 974. iT - 1.41e6T^{2} \)
19 \( 1 - 380.T + 2.47e6T^{2} \)
23 \( 1 - 3.54e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.44e3T + 2.05e7T^{2} \)
31 \( 1 + 3.62e3T + 2.86e7T^{2} \)
37 \( 1 + 1.75e3iT - 6.93e7T^{2} \)
41 \( 1 - 263.T + 1.15e8T^{2} \)
43 \( 1 - 1.44e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.34e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.34e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.90e3T + 7.14e8T^{2} \)
61 \( 1 - 2.94e4T + 8.44e8T^{2} \)
67 \( 1 + 7.16e3iT - 1.35e9T^{2} \)
71 \( 1 + 8.13e4T + 1.80e9T^{2} \)
73 \( 1 - 5.51e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.64e4T + 3.07e9T^{2} \)
83 \( 1 - 1.16e5iT - 3.93e9T^{2} \)
89 \( 1 + 9.93e4T + 5.58e9T^{2} \)
97 \( 1 + 6.29e4iT - 8.58e9T^{2} \)
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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

−15.17045281779094678235287966256, −13.78803870487055344816831536042, −12.74620476992305016464303711538, −11.53075708169948334319772540237, −9.845862134244137508098265373029, −9.065854846696757967028452322982, −7.29719475006371755943081494645, −5.51146339117131121525309030140, −3.98989630035496605159431364145, −1.30247539174827601407890282749, 1.78796556611304294261520414208, 3.96198832993140469169577287452, 6.25915639395993910572028036327, 7.18983649888952279293170130318, 9.080078048263667754726372066457, 10.31815579196641308392877284563, 11.61518941345501370875546222163, 12.88709342614742449051802536162, 14.19456473599661510506845712310, 14.94997884420519436331590234756

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