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Properties

Label 2-280e2-1.1-c1-0-167
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s − 3·11-s − 5·13-s + 3·17-s − 2·19-s − 6·23-s − 5·27-s − 3·29-s − 4·31-s − 3·33-s + 2·37-s − 5·39-s + 12·41-s + 10·43-s − 9·47-s + 3·51-s + 12·53-s − 2·57-s + 8·61-s + 4·67-s − 6·69-s + 2·73-s + 79-s + 81-s + 12·83-s − 3·87-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s − 0.904·11-s − 1.38·13-s + 0.727·17-s − 0.458·19-s − 1.25·23-s − 0.962·27-s − 0.557·29-s − 0.718·31-s − 0.522·33-s + 0.328·37-s − 0.800·39-s + 1.87·41-s + 1.52·43-s − 1.31·47-s + 0.420·51-s + 1.64·53-s − 0.264·57-s + 1.02·61-s + 0.488·67-s − 0.722·69-s + 0.234·73-s + 0.112·79-s + 1/9·81-s + 1.31·83-s − 0.321·87-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: \(1\)
Selberg data: \((2,\ 78400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.33384021656713, −13.92962757583117, −13.18684692741750, −12.85503690553350, −12.32036830040282, −11.84226894464971, −11.29331071220674, −10.69230686874068, −10.24731393451238, −9.627727528963765, −9.320673724580180, −8.701669659243494, −7.980486953113242, −7.754092125111950, −7.369988216426134, −6.543967744659334, −5.822670158098527, −5.490911978921485, −4.920370399951531, −4.118453868852981, −3.671918908380760, −2.859723004327127, −2.333995070876112, −2.056903106799481, −0.7636584697061904, 0, 0.7636584697061904, 2.056903106799481, 2.333995070876112, 2.859723004327127, 3.671918908380760, 4.118453868852981, 4.920370399951531, 5.490911978921485, 5.822670158098527, 6.543967744659334, 7.369988216426134, 7.754092125111950, 7.980486953113242, 8.701669659243494, 9.320673724580180, 9.627727528963765, 10.24731393451238, 10.69230686874068, 11.29331071220674, 11.84226894464971, 12.32036830040282, 12.85503690553350, 13.18684692741750, 13.92962757583117, 14.33384021656713

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