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Properties

Label 2-3200-40.19-c0-0-4
Degree $2$
Conductor $3200$
Sign $-0.447 + 0.894i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s − 1.99·9-s + 1.73·11-s − i·17-s + 1.73·19-s + 1.73i·27-s − 2.99i·33-s − 41-s − 49-s − 1.73·51-s − 2.99i·57-s − 1.73i·67-s + i·73-s + 0.999·81-s + 1.73i·83-s + ⋯
L(s)  = 1  − 1.73i·3-s − 1.99·9-s + 1.73·11-s − i·17-s + 1.73·19-s + 1.73i·27-s − 2.99i·33-s − 41-s − 49-s − 1.73·51-s − 2.99i·57-s − 1.73i·67-s + i·73-s + 0.999·81-s + 1.73i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.365094241\)
\(L(\frac12)\) \(\approx\) \(1.365094241\)
\(L(1)\) 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 \( 1 + 1.73iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - 1.73T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.73iT - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 - 2iT - T^{2} \)
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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

−8.429389338088979554873821931944, −7.72234459521281961781775552239, −6.97505337915531153988296712410, −6.66590090308241233486679925262, −5.79172739906683840547376247573, −4.95784078782762465087232220298, −3.64970143225008491407511945110, −2.81879953781749984825649418514, −1.67225527033260316389962115265, −0.962272015099801944067239710958, 1.48833596958370965274205684930, 3.09812018185450359245456932186, 3.68559860862640257574997850971, 4.33187611646764080216131599842, 5.14298041303352628363515038931, 5.90315604120904633698518190696, 6.68403036771457460349755217084, 7.73521380678105537250780475126, 8.778749000283217275521799396292, 9.056348516839121435878073057219

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