Properties

Label 2-40e2-1.1-c1-0-26
Degree $2$
Conductor $1600$
Sign $-1$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  − 3·9-s − 4·13-s + 8·17-s − 10·29-s − 12·37-s − 10·41-s − 7·49-s + 4·53-s − 10·61-s + 16·73-s + 9·81-s − 10·89-s − 8·97-s + 2·101-s − 6·109-s − 16·113-s + 12·117-s + ⋯
L(s)  = 1  − 9-s − 1.10·13-s + 1.94·17-s − 1.85·29-s − 1.97·37-s − 1.56·41-s − 49-s + 0.549·53-s − 1.28·61-s + 1.87·73-s + 81-s − 1.05·89-s − 0.812·97-s + 0.199·101-s − 0.574·109-s − 1.50·113-s + 1.10·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -1)\)

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 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−9.066578188161773859679921125938, −8.126003367030777913782292064447, −7.52991763396308675946168049663, −6.62707438006867438574420001208, −5.43538712614250917522273729751, −5.22283258119030602960852633346, −3.70896411876379581085476870424, −2.98396765796017704987428125492, −1.73719838151596358495089080076, 0, 1.73719838151596358495089080076, 2.98396765796017704987428125492, 3.70896411876379581085476870424, 5.22283258119030602960852633346, 5.43538712614250917522273729751, 6.62707438006867438574420001208, 7.52991763396308675946168049663, 8.126003367030777913782292064447, 9.066578188161773859679921125938

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