Properties

Label 2-40e2-1.1-c3-0-82
Degree $2$
Conductor $1600$
Sign $-1$
Analytic cond. $94.4030$
Root an. cond. $9.71612$
Motivic weight $3$
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 − 26·7-s − 23·9-s + 28·11-s + 12·13-s + 64·17-s + 60·19-s − 52·21-s + 58·23-s − 100·27-s − 90·29-s − 128·31-s + 56·33-s + 236·37-s + 24·39-s + 242·41-s + 362·43-s − 226·47-s + 333·49-s + 128·51-s − 108·53-s + 120·57-s + 20·59-s − 542·61-s + 598·63-s − 434·67-s + 116·69-s + ⋯
L(s)  = 1  + 0.384·3-s − 1.40·7-s − 0.851·9-s + 0.767·11-s + 0.256·13-s + 0.913·17-s + 0.724·19-s − 0.540·21-s + 0.525·23-s − 0.712·27-s − 0.576·29-s − 0.741·31-s + 0.295·33-s + 1.04·37-s + 0.0985·39-s + 0.921·41-s + 1.28·43-s − 0.701·47-s + 0.970·49-s + 0.351·51-s − 0.279·53-s + 0.278·57-s + 0.0441·59-s − 1.13·61-s + 1.19·63-s − 0.791·67-s + 0.202·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(94.4030\)
Root analytic conductor: \(9.71612\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1600,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 2 T + p^{3} T^{2} \)
7 \( 1 + 26 T + p^{3} T^{2} \)
11 \( 1 - 28 T + p^{3} T^{2} \)
13 \( 1 - 12 T + p^{3} T^{2} \)
17 \( 1 - 64 T + p^{3} T^{2} \)
19 \( 1 - 60 T + p^{3} T^{2} \)
23 \( 1 - 58 T + p^{3} T^{2} \)
29 \( 1 + 90 T + p^{3} T^{2} \)
31 \( 1 + 128 T + p^{3} T^{2} \)
37 \( 1 - 236 T + p^{3} T^{2} \)
41 \( 1 - 242 T + p^{3} T^{2} \)
43 \( 1 - 362 T + p^{3} T^{2} \)
47 \( 1 + 226 T + p^{3} T^{2} \)
53 \( 1 + 108 T + p^{3} T^{2} \)
59 \( 1 - 20 T + p^{3} T^{2} \)
61 \( 1 + 542 T + p^{3} T^{2} \)
67 \( 1 + 434 T + p^{3} T^{2} \)
71 \( 1 + 1128 T + p^{3} T^{2} \)
73 \( 1 + 632 T + p^{3} T^{2} \)
79 \( 1 + 720 T + p^{3} T^{2} \)
83 \( 1 + 478 T + p^{3} T^{2} \)
89 \( 1 + 490 T + p^{3} T^{2} \)
97 \( 1 + 1456 T + p^{3} T^{2} \)
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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

−8.920162829144221099716953829869, −7.78832423478749330062692107840, −7.11102483856514544294828358860, −6.04429690268902801739734924453, −5.68335628732277443719789781289, −4.22330902993462559776836440277, −3.29699920887211990412702646025, −2.80121332076378430714878148003, −1.26257124816117251253609007983, 0, 1.26257124816117251253609007983, 2.80121332076378430714878148003, 3.29699920887211990412702646025, 4.22330902993462559776836440277, 5.68335628732277443719789781289, 6.04429690268902801739734924453, 7.11102483856514544294828358860, 7.78832423478749330062692107840, 8.920162829144221099716953829869

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