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Properties

Label 2-40-8.5-c5-0-18
Degree $2$
Conductor $40$
Sign $-0.820 + 0.571i$
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  + (4.22 − 3.75i)2-s − 25.0i·3-s + (3.78 − 31.7i)4-s + 25i·5-s + (−94.1 − 105. i)6-s + 103.·7-s + (−103. − 148. i)8-s − 384.·9-s + (93.9 + 105. i)10-s + 740. i·11-s + (−796. − 94.7i)12-s − 892. i·13-s + (438. − 389. i)14-s + 626.·15-s + (−995. − 240. i)16-s + 1.13e3·17-s + ⋯
L(s)  = 1  + (0.747 − 0.664i)2-s − 1.60i·3-s + (0.118 − 0.992i)4-s + 0.447i·5-s + (−1.06 − 1.20i)6-s + 0.799·7-s + (−0.571 − 0.820i)8-s − 1.58·9-s + (0.296 + 0.334i)10-s + 1.84i·11-s + (−1.59 − 0.189i)12-s − 1.46i·13-s + (0.597 − 0.530i)14-s + 0.718·15-s + (−0.972 − 0.234i)16-s + 0.955·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.820 + 0.571i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ -0.820 + 0.571i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.710155 - 2.26452i\)
\(L(\frac12)\) \(\approx\) \(0.710155 - 2.26452i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.22 + 3.75i)T \)
5 \( 1 - 25iT \)
good3 \( 1 + 25.0iT - 243T^{2} \)
7 \( 1 - 103.T + 1.68e4T^{2} \)
11 \( 1 - 740. iT - 1.61e5T^{2} \)
13 \( 1 + 892. iT - 3.71e5T^{2} \)
17 \( 1 - 1.13e3T + 1.41e6T^{2} \)
19 \( 1 + 1.15e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.60e3T + 6.43e6T^{2} \)
29 \( 1 - 2.15e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.95e3T + 2.86e7T^{2} \)
37 \( 1 - 4.40e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.78e3T + 1.15e8T^{2} \)
43 \( 1 - 1.30e4iT - 1.47e8T^{2} \)
47 \( 1 + 8.00e3T + 2.29e8T^{2} \)
53 \( 1 + 3.43e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.20e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.82e3iT - 8.44e8T^{2} \)
67 \( 1 - 5.49e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.28e4T + 1.80e9T^{2} \)
73 \( 1 + 2.08e4T + 2.07e9T^{2} \)
79 \( 1 - 3.02e4T + 3.07e9T^{2} \)
83 \( 1 - 9.39e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.17e3T + 5.58e9T^{2} \)
97 \( 1 - 7.41e4T + 8.58e9T^{2} \)
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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

−14.47119336976575317226560504683, −13.15599282552607515430311017225, −12.48411892210895176470009465203, −11.46784092116231564225077832174, −10.07011024416590438853636665883, −7.84501949273412000323688316077, −6.76017320833967476675872180965, −5.11799189552118532592052781462, −2.64642627274454206792165093060, −1.23457771820016463361882648431, 3.54955716641439625793285731279, 4.72358126466787606899155806376, 5.87965806317797403681386465845, 8.190009431663367257003541297932, 9.181397888428049285223917762225, 10.95785166957701712141949821192, 11.87656439932100976283731297774, 13.85584080711270808611455143052, 14.45340836307177643009967417113, 15.70218472151835090392134709066

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