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Properties

Label 2-40-40.19-c2-0-0
Degree $2$
Conductor $40$
Sign $0.402 - 0.915i$
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 + (9.49 + 3.13i)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 + (0.949 + 0.313i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.402 - 0.915i$
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: \((2,\ 40,\ (\ :1),\ 0.402 - 0.915i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.615111 + 0.401473i\)
\(L(\frac12)\) \(\approx\) \(0.615111 + 0.401473i\)
\(L(2)\) 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

−16.08443474108190943143472587327, −15.31894439949706755262417439472, −14.05381441387147076749587476821, −11.87273571401558725884927839531, −11.11174454943616752340380696667, −10.24356717857660126308414241403, −8.913937521577547923370400116895, −7.71757193569489661107274400862, −4.71249889791917625667617331848, −3.41511020684141249973802212171, 1.23639856980045996435091134825, 5.28127796982565570317723860519, 7.09228584183052193968208014755, 7.85305766796345337078923135365, 8.821876377136103019972685289977, 11.10590535702895958782318112986, 12.10568556585895037945042833463, 13.53049715497168707439373352918, 14.55525098762765270852735776414, 15.87007223813240618475237189025

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