Properties

Label 2-280e2-1.1-c1-0-0
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  + 2·3-s + 9-s − 5·11-s − 8·17-s − 2·19-s − 7·23-s − 4·27-s + 3·29-s − 4·31-s − 10·33-s + 37-s + 2·41-s − 3·43-s − 6·47-s − 16·51-s − 10·53-s − 4·57-s − 4·59-s − 6·61-s − 13·67-s − 14·69-s + 5·71-s − 6·73-s − 13·79-s − 11·81-s + 16·83-s + 6·87-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 1.50·11-s − 1.94·17-s − 0.458·19-s − 1.45·23-s − 0.769·27-s + 0.557·29-s − 0.718·31-s − 1.74·33-s + 0.164·37-s + 0.312·41-s − 0.457·43-s − 0.875·47-s − 2.24·51-s − 1.37·53-s − 0.529·57-s − 0.520·59-s − 0.768·61-s − 1.58·67-s − 1.68·69-s + 0.593·71-s − 0.702·73-s − 1.46·79-s − 1.22·81-s + 1.75·83-s + 0.643·87-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

\(L(1)\) \(\approx\) \(0.3220017412\)
\(L(\frac12)\) \(\approx\) \(0.3220017412\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.97887256961283, −13.47637252691406, −13.19382446800679, −12.75131574849977, −12.11735187291923, −11.45662249970946, −10.90671680996978, −10.55006685276689, −9.914601274393199, −9.437387137118557, −8.772911202372704, −8.553103906264655, −7.789404102855903, −7.746192927645264, −6.900148852500808, −6.237106067669335, −5.832629607862472, −4.928602788201322, −4.534536755946318, −3.916024800839035, −3.160852699294151, −2.695586266619348, −2.101451831957016, −1.700455586612192, −0.1526439312750572, 0.1526439312750572, 1.700455586612192, 2.101451831957016, 2.695586266619348, 3.160852699294151, 3.916024800839035, 4.534536755946318, 4.928602788201322, 5.832629607862472, 6.237106067669335, 6.900148852500808, 7.746192927645264, 7.789404102855903, 8.553103906264655, 8.772911202372704, 9.437387137118557, 9.914601274393199, 10.55006685276689, 10.90671680996978, 11.45662249970946, 12.11735187291923, 12.75131574849977, 13.19382446800679, 13.47637252691406, 13.97887256961283

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