Properties

Label 2-40e2-1.1-c3-0-28
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  − 27·9-s + 92·13-s + 104·17-s − 130·29-s − 396·37-s + 230·41-s − 343·49-s − 572·53-s + 830·61-s + 592·73-s + 729·81-s + 1.67e3·89-s + 1.81e3·97-s − 598·101-s + 1.74e3·109-s + 1.32e3·113-s − 2.48e3·117-s + ⋯
L(s)  = 1  − 9-s + 1.96·13-s + 1.48·17-s − 0.832·29-s − 1.75·37-s + 0.876·41-s − 49-s − 1.48·53-s + 1.74·61-s + 0.949·73-s + 81-s + 1.98·89-s + 1.90·97-s − 0.589·101-s + 1.53·109-s + 1.10·113-s − 1.96·117-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: \(0\)
Selberg data: \((2,\ 1600,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.170747332\)
\(L(\frac12)\) \(\approx\) \(2.170747332\)
\(L(\frac{5}{2})\) 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 \( 1 + p^{3} T^{2} \)
7 \( 1 + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 - 92 T + p^{3} T^{2} \)
17 \( 1 - 104 T + p^{3} T^{2} \)
19 \( 1 + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + 130 T + p^{3} T^{2} \)
31 \( 1 + p^{3} T^{2} \)
37 \( 1 + 396 T + p^{3} T^{2} \)
41 \( 1 - 230 T + p^{3} T^{2} \)
43 \( 1 + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + 572 T + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 - 830 T + p^{3} T^{2} \)
67 \( 1 + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 - 592 T + p^{3} T^{2} \)
79 \( 1 + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 - 1670 T + p^{3} T^{2} \)
97 \( 1 - 1816 T + p^{3} T^{2} \)
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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

−8.910638815477927300418543725229, −8.322731025003355346338596659987, −7.60334988348691863282311097380, −6.43705211097272875357808887672, −5.82621934776736941704139075647, −5.09289003427586174508546352719, −3.67128040244791292349057637806, −3.26210713818336384030711292775, −1.82182526404571793481162085201, −0.72776802490996063526525155967, 0.72776802490996063526525155967, 1.82182526404571793481162085201, 3.26210713818336384030711292775, 3.67128040244791292349057637806, 5.09289003427586174508546352719, 5.82621934776736941704139075647, 6.43705211097272875357808887672, 7.60334988348691863282311097380, 8.322731025003355346338596659987, 8.910638815477927300418543725229

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