Properties

Label 2-40e2-1.1-c3-0-10
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 $0$

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: $\chi_{1600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8150365103\)
\(L(\frac12)\) \(\approx\) \(0.8150365103\)
\(L(\frac{5}{2})\) 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

−9.091664289665640065437549244409, −8.360645285530265661178043368799, −7.45957465379418827471691747938, −6.56565085334744627512392232436, −5.92635029375366982747429211298, −4.92510337604195680295580437135, −3.79249977571853771674370656849, −2.90879686690684928297338092565, −2.25606953382465632073228379385, −0.39426574137207148776708072076, 0.39426574137207148776708072076, 2.25606953382465632073228379385, 2.90879686690684928297338092565, 3.79249977571853771674370656849, 4.92510337604195680295580437135, 5.92635029375366982747429211298, 6.56565085334744627512392232436, 7.45957465379418827471691747938, 8.360645285530265661178043368799, 9.091664289665640065437549244409

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