Properties

Label 2-40e2-8.3-c2-0-67
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  − 5.13·3-s − 6.19i·7-s + 17.3·9-s − 20.0·11-s − 15.8i·13-s − 6.98·17-s − 10.3·19-s + 31.7i·21-s − 22.3i·23-s − 42.9·27-s + 4.20i·29-s − 20.7i·31-s + 102.·33-s − 35.4i·37-s + 81.4i·39-s + ⋯
L(s)  = 1  − 1.71·3-s − 0.884i·7-s + 1.92·9-s − 1.82·11-s − 1.22i·13-s − 0.411·17-s − 0.546·19-s + 1.51i·21-s − 0.973i·23-s − 1.58·27-s + 0.145i·29-s − 0.668i·31-s + 3.11·33-s − 0.958i·37-s + 2.08i·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

\(L(\frac{3}{2})\) \(\approx\) \(0.1413440328\)
\(L(\frac12)\) \(\approx\) \(0.1413440328\)
\(L(2)\) 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 \( 1 + 5.13T + 9T^{2} \)
7 \( 1 + 6.19iT - 49T^{2} \)
11 \( 1 + 20.0T + 121T^{2} \)
13 \( 1 + 15.8iT - 169T^{2} \)
17 \( 1 + 6.98T + 289T^{2} \)
19 \( 1 + 10.3T + 361T^{2} \)
23 \( 1 + 22.3iT - 529T^{2} \)
29 \( 1 - 4.20iT - 841T^{2} \)
31 \( 1 + 20.7iT - 961T^{2} \)
37 \( 1 + 35.4iT - 1.36e3T^{2} \)
41 \( 1 + 37.0T + 1.68e3T^{2} \)
43 \( 1 + 23.8T + 1.84e3T^{2} \)
47 \( 1 + 48.7iT - 2.20e3T^{2} \)
53 \( 1 + 77.4iT - 2.80e3T^{2} \)
59 \( 1 - 0.497T + 3.48e3T^{2} \)
61 \( 1 + 60.7iT - 3.72e3T^{2} \)
67 \( 1 - 82.5T + 4.48e3T^{2} \)
71 \( 1 - 28.7iT - 5.04e3T^{2} \)
73 \( 1 + 10.1T + 5.32e3T^{2} \)
79 \( 1 + 87.5iT - 6.24e3T^{2} \)
83 \( 1 + 103.T + 6.88e3T^{2} \)
89 \( 1 - 49.2T + 7.92e3T^{2} \)
97 \( 1 - 84.4T + 9.40e3T^{2} \)
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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

−8.459993012012192111655788480540, −7.69384877764693466212380190709, −6.93850809712419800863270960296, −6.13222374979797803730175278307, −5.25450184323625152062723472499, −4.85154603105226105116897956479, −3.70744936210308129550374804935, −2.28522620874463886430385525388, −0.60875984293450682251393250313, −0.084244231930359354787548293058, 1.56989749936602300936967022802, 2.72715898309472530797392218185, 4.30259104226418500716396286785, 5.08471843142067083651071836084, 5.61446687951672852041994826124, 6.42067317163024816724204707551, 7.14271432752261256102182189073, 8.134515085107643800004689707653, 9.086322761061004118307319826122, 10.02038277821242003742672681148

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