Properties

Label 2-80e2-1.1-c1-0-115
Degree $2$
Conductor $6400$
Sign $-1$
Analytic cond. $51.1042$
Root an. cond. $7.14872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.732·3-s − 1.26·7-s − 2.46·9-s + 3.46·11-s + 3.46·13-s − 3.46·17-s − 2·19-s − 0.928·21-s − 8.19·23-s − 4·27-s + 9.46·31-s + 2.53·33-s + 6·37-s + 2.53·39-s − 2.53·41-s + 10.1·43-s − 8.19·47-s − 5.39·49-s − 2.53·51-s − 10.3·53-s − 1.46·57-s − 6·59-s + 12.9·61-s + 3.12·63-s − 10.1·67-s − 6·69-s − 4.39·71-s + ⋯
L(s)  = 1  + 0.422·3-s − 0.479·7-s − 0.821·9-s + 1.04·11-s + 0.960·13-s − 0.840·17-s − 0.458·19-s − 0.202·21-s − 1.70·23-s − 0.769·27-s + 1.69·31-s + 0.441·33-s + 0.986·37-s + 0.406·39-s − 0.396·41-s + 1.55·43-s − 1.19·47-s − 0.770·49-s − 0.355·51-s − 1.42·53-s − 0.193·57-s − 0.781·59-s + 1.65·61-s + 0.393·63-s − 1.24·67-s − 0.722·69-s − 0.521·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(51.1042\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ -1)\)

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 \( 1 - 0.732T + 3T^{2} \)
7 \( 1 + 1.26T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 8.19T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 9.46T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + 8.19T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + 4.39T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 + 0.928T + 89T^{2} \)
97 \( 1 - 6.39T + 97T^{2} \)
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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

−8.019940126157620406677920982256, −6.75803191026166150750464997870, −6.23818783402361166452080925409, −5.86740213865480583927191965400, −4.54079186302508625837961444503, −4.00511038198653912498315091926, −3.18964498554947945215223205728, −2.39739420280263890402705029285, −1.38383270581844470717905273173, 0, 1.38383270581844470717905273173, 2.39739420280263890402705029285, 3.18964498554947945215223205728, 4.00511038198653912498315091926, 4.54079186302508625837961444503, 5.86740213865480583927191965400, 6.23818783402361166452080925409, 6.75803191026166150750464997870, 8.019940126157620406677920982256

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