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Properties

Label 2-40-40.29-c1-0-0
Degree $2$
Conductor $40$
Sign $0.632 - 0.774i$
Analytic cond. $0.319401$
Root an. cond. $0.565156$
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  + (−0.707 + 1.22i)2-s + 1.41·3-s + (−0.999 − 1.73i)4-s + (−1.41 + 1.73i)5-s + (−1.00 + 1.73i)6-s − 2.44i·7-s + 2.82·8-s − 0.999·9-s + (−1.12 − 2.95i)10-s − 3.46i·11-s + (−1.41 − 2.44i)12-s + (2.99 + 1.73i)14-s + (−2.00 + 2.44i)15-s + (−2.00 + 3.46i)16-s + 4.89i·17-s + (0.707 − 1.22i)18-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)2-s + 0.816·3-s + (−0.499 − 0.866i)4-s + (−0.632 + 0.774i)5-s + (−0.408 + 0.707i)6-s − 0.925i·7-s + 0.999·8-s − 0.333·9-s + (−0.354 − 0.935i)10-s − 1.04i·11-s + (−0.408 − 0.707i)12-s + (0.801 + 0.462i)14-s + (−0.516 + 0.632i)15-s + (−0.500 + 0.866i)16-s + 1.18i·17-s + (0.166 − 0.288i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.632 - 0.774i$
Analytic conductor: \(0.319401\)
Root analytic conductor: \(0.565156\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1/2),\ 0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613907 + 0.291297i\)
\(L(\frac12)\) \(\approx\) \(0.613907 + 0.291297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
5 \( 1 + (1.41 - 1.73i)T \)
good3 \( 1 - 1.41T + 3T^{2} \)
7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 2.44iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 + 7.34iT - 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 4.24T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−16.35473157101406880141352507957, −15.10410071283788008354361173156, −14.33056500355143541254288766920, −13.47805915299557003078475780060, −11.21121993960389503000334126595, −10.11235888394124421199352608016, −8.461263238288824085470361171638, −7.70014586697367922220762333574, −6.20129881988782172981086432450, −3.73298347191104287828049766918, 2.68614116154331665113960259570, 4.67560823073527008864014994875, 7.66792177713269141433414264714, 8.798554870288155207523017078000, 9.528438312547950137029041324522, 11.45864479324728532217164289160, 12.30646849888044221845573656202, 13.43132523446325291646408089333, 14.91739322937852384045281507519, 16.07741453009648080787786585230

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