Properties

Label 2-3200-40.29-c1-0-5
Degree $2$
Conductor $3200$
Sign $-0.948 + 0.316i$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4i·7-s − 2·9-s + 3i·11-s i·17-s + 7i·19-s − 4i·21-s − 4i·23-s + 5·27-s + 8i·29-s − 4·31-s − 3i·33-s − 4·37-s + 3·41-s + 8·43-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51i·7-s − 0.666·9-s + 0.904i·11-s − 0.242i·17-s + 1.60i·19-s − 0.872i·21-s − 0.834i·23-s + 0.962·27-s + 1.48i·29-s − 0.718·31-s − 0.522i·33-s − 0.657·37-s + 0.468·41-s + 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.948 + 0.316i$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ -0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5154983854\)
\(L(\frac12)\) \(\approx\) \(0.5154983854\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 3T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + iT - 17T^{2} \)
19 \( 1 - 7iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + 8iT - 59T^{2} \)
61 \( 1 - 4iT - 61T^{2} \)
67 \( 1 + 9T + 67T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + 13T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−9.049560697432852885851378854964, −8.421765471740855846526678920095, −7.64015623870181745785009872076, −6.63591557567047728067448045055, −5.95945423322715080601183065980, −5.36802178796702621293015918707, −4.71481752233690790434291693863, −3.47664571005465371696842967054, −2.54203858623779619495287386620, −1.65085841388583397591508596478, 0.19358975436772309142960306434, 1.05773738251783684196571010817, 2.61431931021041846750722624654, 3.57703563113304929716118509773, 4.34613104888088534349692802905, 5.24255833252512168199715169886, 6.00375224817096469608045648535, 6.72702726690258594056126029637, 7.46810545084120630596574467585, 8.150857942610757783595936539909

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