Properties

Label 2-80e2-1.1-c1-0-75
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·3-s + 5.00·9-s + 2.82·11-s − 6·17-s + 8.48·19-s + 5.65·27-s + 8.00·33-s + 6·41-s + 8.48·43-s − 7·49-s − 16.9·51-s + 24·57-s + 14.1·59-s − 8.48·67-s − 2·73-s + 1.00·81-s + 2.82·83-s + 18·89-s + 10·97-s + 14.1·99-s + 19.7·107-s − 18·113-s + ⋯
L(s)  = 1  + 1.63·3-s + 1.66·9-s + 0.852·11-s − 1.45·17-s + 1.94·19-s + 1.08·27-s + 1.39·33-s + 0.937·41-s + 1.29·43-s − 49-s − 2.37·51-s + 3.17·57-s + 1.84·59-s − 1.03·67-s − 0.234·73-s + 0.111·81-s + 0.310·83-s + 1.90·89-s + 1.01·97-s + 1.42·99-s + 1.91·107-s − 1.69·113-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: \(0\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.225655767\)
\(L(\frac12)\) \(\approx\) \(4.225655767\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 8.48T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.004604059297186575675466799584, −7.47739101701819609266179145244, −6.86190371842249173263160904110, −6.02842313016901507347653475403, −4.96673371219222678923271082352, −4.15213607452696720572799923352, −3.54584521107838647336478443335, −2.77015956688253651923819653440, −2.03809384289219825909719242311, −1.04057642387582761234161837251, 1.04057642387582761234161837251, 2.03809384289219825909719242311, 2.77015956688253651923819653440, 3.54584521107838647336478443335, 4.15213607452696720572799923352, 4.96673371219222678923271082352, 6.02842313016901507347653475403, 6.86190371842249173263160904110, 7.47739101701819609266179145244, 8.004604059297186575675466799584

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