Properties

Label 2-280e2-1.1-c1-0-124
Degree $2$
Conductor $78400$
Sign $1$
Analytic cond. $626.027$
Root an. cond. $25.0205$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 6·11-s + 4·13-s + 7·17-s − 4·19-s − 23-s + 6·29-s + 3·31-s + 4·37-s + 9·41-s − 8·43-s − 11·47-s + 4·53-s − 10·59-s + 2·61-s − 8·67-s − 3·71-s + 2·73-s − 17·79-s + 9·81-s + 2·83-s + 7·89-s − 97-s − 18·99-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 9-s + 1.80·11-s + 1.10·13-s + 1.69·17-s − 0.917·19-s − 0.208·23-s + 1.11·29-s + 0.538·31-s + 0.657·37-s + 1.40·41-s − 1.21·43-s − 1.60·47-s + 0.549·53-s − 1.30·59-s + 0.256·61-s − 0.977·67-s − 0.356·71-s + 0.234·73-s − 1.91·79-s + 81-s + 0.219·83-s + 0.741·89-s − 0.101·97-s − 1.80·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(626.027\)
Root analytic conductor: \(25.0205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 78400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.175031109\)
\(L(\frac12)\) \(\approx\) \(3.175031109\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.93614960084548, −13.85926947868646, −12.88589474840124, −12.56406621570600, −11.86583230250079, −11.51180368829188, −11.30604209526617, −10.37132479221496, −10.12369006476040, −9.397560397002452, −8.871816409685668, −8.535554926173630, −8.001057935885731, −7.453480857955286, −6.509833373403232, −6.293440095899989, −5.931007925414550, −5.183676009097048, −4.412123192655633, −3.978497304662167, −3.222710517310009, −2.969457288775401, −1.856186357125840, −1.270889603558052, −0.6391274117039518, 0.6391274117039518, 1.270889603558052, 1.856186357125840, 2.969457288775401, 3.222710517310009, 3.978497304662167, 4.412123192655633, 5.183676009097048, 5.931007925414550, 6.293440095899989, 6.509833373403232, 7.453480857955286, 8.001057935885731, 8.535554926173630, 8.871816409685668, 9.397560397002452, 10.12369006476040, 10.37132479221496, 11.30604209526617, 11.51180368829188, 11.86583230250079, 12.56406621570600, 12.88589474840124, 13.85926947868646, 13.93614960084548

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