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Properties

Label 2-40-40.19-c2-0-4
Degree $2$
Conductor $40$
Sign $0.991 + 0.127i$
Analytic cond. $1.08992$
Root an. cond. $1.04399$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 − 1.48i)2-s + 4.79i·3-s + (−0.381 − 3.98i)4-s + (4.35 − 2.46i)5-s + (7.09 + 6.44i)6-s − 7.67·7-s + (−6.40 − 4.79i)8-s − 13.9·9-s + (2.21 − 9.75i)10-s − 0.472·11-s + (19.0 − 1.82i)12-s − 4.10·13-s + (−10.3 + 11.3i)14-s + (11.7 + 20.8i)15-s + (−15.7 + 3.04i)16-s + 2.26i·17-s + ⋯
L(s)  = 1  + (0.672 − 0.740i)2-s + 1.59i·3-s + (−0.0954 − 0.995i)4-s + (0.870 − 0.492i)5-s + (1.18 + 1.07i)6-s − 1.09·7-s + (−0.800 − 0.598i)8-s − 1.54·9-s + (0.221 − 0.975i)10-s − 0.0429·11-s + (1.58 − 0.152i)12-s − 0.316·13-s + (−0.737 + 0.811i)14-s + (0.785 + 1.38i)15-s + (−0.981 + 0.190i)16-s + 0.133i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.991 + 0.127i$
Analytic conductor: \(1.08992\)
Root analytic conductor: \(1.04399\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.41744 - 0.0903881i\)
\(L(\frac12)\) \(\approx\) \(1.41744 - 0.0903881i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 1.48i)T \)
5 \( 1 + (-4.35 + 2.46i)T \)
good3 \( 1 - 4.79iT - 9T^{2} \)
7 \( 1 + 7.67T + 49T^{2} \)
11 \( 1 + 0.472T + 121T^{2} \)
13 \( 1 + 4.10T + 169T^{2} \)
17 \( 1 - 2.26iT - 289T^{2} \)
19 \( 1 - 26.3T + 361T^{2} \)
23 \( 1 + 9.73T + 529T^{2} \)
29 \( 1 - 41.6iT - 841T^{2} \)
31 \( 1 + 22.0iT - 961T^{2} \)
37 \( 1 - 51.7T + 1.36e3T^{2} \)
41 \( 1 - 15.0T + 1.68e3T^{2} \)
43 \( 1 - 9.31iT - 1.84e3T^{2} \)
47 \( 1 + 8.76T + 2.20e3T^{2} \)
53 \( 1 + 39.9T + 2.80e3T^{2} \)
59 \( 1 + 77.1T + 3.48e3T^{2} \)
61 \( 1 - 14.7iT - 3.72e3T^{2} \)
67 \( 1 + 75.8iT - 4.48e3T^{2} \)
71 \( 1 + 81.0iT - 5.04e3T^{2} \)
73 \( 1 + 83.4iT - 5.32e3T^{2} \)
79 \( 1 - 100. iT - 6.24e3T^{2} \)
83 \( 1 + 0.266iT - 6.88e3T^{2} \)
89 \( 1 - 85.5T + 7.92e3T^{2} \)
97 \( 1 + 99.1iT - 9.40e3T^{2} \)
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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

−15.89603994628567490189009827728, −14.68418054510285600597617200925, −13.58740535556211882441918002574, −12.39665103562780619664396799025, −10.85211666274817414189161616856, −9.710891809630203963534704267860, −9.375896654509953681484259389007, −5.97363015588864742175782809166, −4.78123612387522035208309428418, −3.20405183829025395931864230915, 2.79087151347192626642113680792, 5.82711337425286120671182137750, 6.69630817025611488811321182641, 7.72185760346141899752556872776, 9.546840770480893704870511763237, 11.76243183641693071055843758583, 12.86869802441875309338322080888, 13.54447436532303047357732025562, 14.37749540679169841636589348131, 15.95962451268665410928672854258

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