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Properties

Label 2-40e2-8.3-c2-0-44
Degree $2$
Conductor $1600$
Sign $0.965 + 0.258i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.965 + 0.258i$
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: \((2,\ 1600,\ (\ :1),\ 0.965 + 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.308748538\)
\(L(\frac12)\) \(\approx\) \(2.308748538\)
\(L(2)\) 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

−9.154185966116779338939320714057, −8.150735732022759207071484870362, −8.034442593388773856300391082497, −6.82007865428910424297615203527, −5.76822614559639554052566676853, −5.24171586609633589146848118760, −3.98712981723430187240380748142, −2.91183005124181815451955121778, −2.39694750782310509007797620491, −0.73950484357919577143123713528, 0.925051751962038654062793569838, 2.25337347460548699207429105098, 3.29955533530446005953413210839, 3.93305062712498768932682772124, 5.31766315494277626475284983545, 5.78260464469633010679871479953, 7.20964265534863721703776066250, 7.60780061758701801146712912796, 8.408440570964492380104803260631, 9.331806685248574528108692429820

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