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Properties

Label 2-40e2-20.19-c2-0-4
Degree $2$
Conductor $1600$
Sign $-0.894 - 0.447i$
Analytic cond. $43.5968$
Root an. cond. $6.60279$
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  + 3.87·3-s − 7.74·7-s + 6.00·9-s − 19.3i·11-s + 20i·13-s + 15i·17-s + 19.3i·19-s − 30.0·21-s − 7.74·23-s − 11.6·27-s − 48·29-s − 38.7i·31-s − 75i·33-s − 10i·37-s + 77.4i·39-s + ⋯
L(s)  = 1  + 1.29·3-s − 1.10·7-s + 0.666·9-s − 1.76i·11-s + 1.53i·13-s + 0.882i·17-s + 1.01i·19-s − 1.42·21-s − 0.336·23-s − 0.430·27-s − 1.65·29-s − 1.24i·31-s − 2.27i·33-s − 0.270i·37-s + 1.98i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(43.5968\)
Root analytic conductor: \(6.60279\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1),\ -0.894 - 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6083664650\)
\(L(\frac12)\) \(\approx\) \(0.6083664650\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 3.87T + 9T^{2} \)
7 \( 1 + 7.74T + 49T^{2} \)
11 \( 1 + 19.3iT - 121T^{2} \)
13 \( 1 - 20iT - 169T^{2} \)
17 \( 1 - 15iT - 289T^{2} \)
19 \( 1 - 19.3iT - 361T^{2} \)
23 \( 1 + 7.74T + 529T^{2} \)
29 \( 1 + 48T + 841T^{2} \)
31 \( 1 + 38.7iT - 961T^{2} \)
37 \( 1 + 10iT - 1.36e3T^{2} \)
41 \( 1 - 33T + 1.68e3T^{2} \)
43 \( 1 + 61.9T + 1.84e3T^{2} \)
47 \( 1 - 15.4T + 2.20e3T^{2} \)
53 \( 1 - 30iT - 2.80e3T^{2} \)
59 \( 1 - 77.4iT - 3.48e3T^{2} \)
61 \( 1 + 38T + 3.72e3T^{2} \)
67 \( 1 + 58.0T + 4.48e3T^{2} \)
71 \( 1 - 77.4iT - 5.04e3T^{2} \)
73 \( 1 + 5iT - 5.32e3T^{2} \)
79 \( 1 + 38.7iT - 6.24e3T^{2} \)
83 \( 1 - 11.6T + 6.88e3T^{2} \)
89 \( 1 + 87T + 7.92e3T^{2} \)
97 \( 1 - 110iT - 9.40e3T^{2} \)
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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

−9.316247679724688766745707698721, −8.858603080913722829354857208460, −8.130551240425621280368848770073, −7.34363046648085072475525745910, −6.18977039868320919050752509999, −5.83492115815163910189740505673, −3.96892353268548356174172564752, −3.68630032555508825848437707682, −2.70643464575483640511428446254, −1.65057495812841922936683716849, 0.12247708277165104184218534983, 1.90890127065785973603918165216, 2.89605907136167870630389803603, 3.40126545421510585696796458976, 4.59869851624030816619910812754, 5.51734551875569669528257526177, 6.78875473867402231960887845908, 7.34772009702242467531969674833, 8.061280246157577280274644819755, 9.028005797837087965308299419641

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