Properties

Label 2-80e2-1.1-c1-0-134
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  + 2.23·3-s + 2.00·9-s − 2.23·11-s − 4·13-s + 3·17-s − 2.23·19-s + 8.94·23-s − 2.23·27-s − 4·29-s − 8.94·31-s − 5.00·33-s − 8·37-s − 8.94·39-s + 5·41-s − 8.94·43-s − 8.94·47-s − 7·49-s + 6.70·51-s − 4·53-s − 5.00·57-s + 8.94·59-s − 8·61-s + 6.70·67-s + 20.0·69-s − 8.94·71-s + 9·73-s − 11·81-s + ⋯
L(s)  = 1  + 1.29·3-s + 0.666·9-s − 0.674·11-s − 1.10·13-s + 0.727·17-s − 0.512·19-s + 1.86·23-s − 0.430·27-s − 0.742·29-s − 1.60·31-s − 0.870·33-s − 1.31·37-s − 1.43·39-s + 0.780·41-s − 1.36·43-s − 1.30·47-s − 49-s + 0.939·51-s − 0.549·53-s − 0.662·57-s + 1.16·59-s − 1.02·61-s + 0.819·67-s + 2.40·69-s − 1.06·71-s + 1.05·73-s − 1.22·81-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 \( 1 - 2.23T + 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 2.23T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 2.23T + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 8.94T + 43T^{2} \)
47 \( 1 + 8.94T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 6.70T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−7.62630228742111104405410194476, −7.31567032450017808789058034206, −6.42579197955964272677006762392, −5.20261900503575326851900162906, −4.98615117825145690684154539801, −3.66526982431319312576559928492, −3.20993716958643536365600947645, −2.40042801312707350751029567853, −1.63557109521418909610674569274, 0, 1.63557109521418909610674569274, 2.40042801312707350751029567853, 3.20993716958643536365600947645, 3.66526982431319312576559928492, 4.98615117825145690684154539801, 5.20261900503575326851900162906, 6.42579197955964272677006762392, 7.31567032450017808789058034206, 7.62630228742111104405410194476

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