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Properties

Label 2-40e2-8.3-c2-0-50
Degree $2$
Conductor $1600$
Sign $-0.258 + 0.965i$
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  − 1.90·3-s − 3.95i·7-s − 5.35·9-s + 6.18·11-s − 4.94i·13-s + 22.4·17-s − 10.3·19-s + 7.54i·21-s + 39.6i·23-s + 27.4·27-s − 30.4i·29-s − 24.7i·31-s − 11.8·33-s + 24.0i·37-s + 9.43i·39-s + ⋯
L(s)  = 1  − 0.636·3-s − 0.565i·7-s − 0.595·9-s + 0.562·11-s − 0.380i·13-s + 1.31·17-s − 0.546·19-s + 0.359i·21-s + 1.72i·23-s + 1.01·27-s − 1.04i·29-s − 0.797i·31-s − 0.357·33-s + 0.649i·37-s + 0.241i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(43.5968\)
Root analytic conductor: \(6.60279\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1),\ -0.258 + 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.001660149\)
\(L(\frac12)\) \(\approx\) \(1.001660149\)
\(L(2)\) 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 \( 1 + 1.90T + 9T^{2} \)
7 \( 1 + 3.95iT - 49T^{2} \)
11 \( 1 - 6.18T + 121T^{2} \)
13 \( 1 + 4.94iT - 169T^{2} \)
17 \( 1 - 22.4T + 289T^{2} \)
19 \( 1 + 10.3T + 361T^{2} \)
23 \( 1 - 39.6iT - 529T^{2} \)
29 \( 1 + 30.4iT - 841T^{2} \)
31 \( 1 + 24.7iT - 961T^{2} \)
37 \( 1 - 24.0iT - 1.36e3T^{2} \)
41 \( 1 - 31.0T + 1.68e3T^{2} \)
43 \( 1 + 52.2T + 1.84e3T^{2} \)
47 \( 1 + 13.1iT - 2.20e3T^{2} \)
53 \( 1 - 17.9iT - 2.80e3T^{2} \)
59 \( 1 + 104.T + 3.48e3T^{2} \)
61 \( 1 + 57.2iT - 3.72e3T^{2} \)
67 \( 1 + 99.3T + 4.48e3T^{2} \)
71 \( 1 - 16.7iT - 5.04e3T^{2} \)
73 \( 1 - 96.3T + 5.32e3T^{2} \)
79 \( 1 + 139. iT - 6.24e3T^{2} \)
83 \( 1 - 91.7T + 6.88e3T^{2} \)
89 \( 1 - 94.7T + 7.92e3T^{2} \)
97 \( 1 + 143.T + 9.40e3T^{2} \)
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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

−9.097501925533590940922052050823, −7.999895069387056612932385285554, −7.50891495460651752210841542490, −6.35433142332606089848209921076, −5.81440136634431034439500310163, −4.95301113281107075033075239571, −3.88367268765474793603055851712, −3.02844215842964229392417982114, −1.50955799486182295585550676321, −0.34824319853303949863942364080, 1.07014026919670234862884273822, 2.43615848458544421354620861312, 3.45703582715618883217917807300, 4.63329625958482473962316924258, 5.41258073570362022901487982061, 6.21335287381003305799474839030, 6.80715000835295631091790775991, 7.954929738769122421449056539653, 8.753659208087697580394005050346, 9.295704743113331509337813423643

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