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Properties

Label 2-3200-1.1-c1-0-42
Degree $2$
Conductor $3200$
Sign $-1$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s + 2·11-s − 2·13-s − 6·17-s + 6·19-s + 4·27-s + 10·29-s − 8·31-s − 4·33-s − 2·37-s + 4·39-s − 6·41-s + 2·43-s + 12·47-s − 7·49-s + 12·51-s − 10·53-s − 12·57-s − 6·59-s − 6·61-s + 14·67-s − 4·71-s + 10·73-s + 8·79-s − 11·81-s − 10·83-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.45·17-s + 1.37·19-s + 0.769·27-s + 1.85·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.304·43-s + 1.75·47-s − 49-s + 1.68·51-s − 1.37·53-s − 1.58·57-s − 0.781·59-s − 0.768·61-s + 1.71·67-s − 0.474·71-s + 1.17·73-s + 0.900·79-s − 1.22·81-s − 1.09·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.319488359616038251493216779987, −7.31441808541618805237936368479, −6.67578829026404213813494591990, −6.09619232320204608553314630106, −5.13298624882772498714676294626, −4.71800197815524482666005499148, −3.61711721453294155791766586903, −2.52254249000789419447963924921, −1.23931743759910414581930506441, 0, 1.23931743759910414581930506441, 2.52254249000789419447963924921, 3.61711721453294155791766586903, 4.71800197815524482666005499148, 5.13298624882772498714676294626, 6.09619232320204608553314630106, 6.67578829026404213813494591990, 7.31441808541618805237936368479, 8.319488359616038251493216779987

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