Properties

Label 2-40e2-4.3-c2-0-34
Degree $2$
Conductor $1600$
Sign $1$
Analytic cond. $43.5968$
Root an. cond. $6.60279$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·9-s − 24·13-s + 16·17-s + 42·29-s + 24·37-s − 18·41-s + 49·49-s + 56·53-s − 22·61-s − 96·73-s + 81·81-s + 78·89-s − 144·97-s + 198·101-s + 182·109-s + 224·113-s − 216·117-s + ⋯
L(s)  = 1  + 9-s − 1.84·13-s + 0.941·17-s + 1.44·29-s + 0.648·37-s − 0.439·41-s + 49-s + 1.05·53-s − 0.360·61-s − 1.31·73-s + 81-s + 0.876·89-s − 1.48·97-s + 1.96·101-s + 1.66·109-s + 1.98·113-s − 1.84·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(43.5968\)
Root analytic conductor: \(6.60279\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1600} (1151, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.020032969\)
\(L(\frac12)\) \(\approx\) \(2.020032969\)
\(L(2)\) 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 \( ( 1 - p T )( 1 + p T ) \)
7 \( ( 1 - p T )( 1 + p T ) \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 + 24 T + p^{2} T^{2} \)
17 \( 1 - 16 T + p^{2} T^{2} \)
19 \( ( 1 - p T )( 1 + p T ) \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( 1 - 42 T + p^{2} T^{2} \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( 1 - 24 T + p^{2} T^{2} \)
41 \( 1 + 18 T + p^{2} T^{2} \)
43 \( ( 1 - p T )( 1 + p T ) \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( 1 - 56 T + p^{2} T^{2} \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 + 22 T + p^{2} T^{2} \)
67 \( ( 1 - p T )( 1 + p T ) \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 + 96 T + p^{2} T^{2} \)
79 \( ( 1 - p T )( 1 + p T ) \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( 1 - 78 T + p^{2} T^{2} \)
97 \( 1 + 144 T + p^{2} T^{2} \)
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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

−9.393042814151218165091305975053, −8.367947437203732592681296377908, −7.44135337971208069411033871903, −7.06906256904773512946803028043, −5.95531746637931879285004755844, −4.94864500131543919239892957295, −4.34684842009753130179988061886, −3.12336075728947567052362166040, −2.11480099809242131745707964976, −0.805396409916092753756013984338, 0.805396409916092753756013984338, 2.11480099809242131745707964976, 3.12336075728947567052362166040, 4.34684842009753130179988061886, 4.94864500131543919239892957295, 5.95531746637931879285004755844, 7.06906256904773512946803028043, 7.44135337971208069411033871903, 8.367947437203732592681296377908, 9.393042814151218165091305975053

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