Properties

Label 2-64400-1.1-c1-0-36
Degree $2$
Conductor $64400$
Sign $-1$
Analytic cond. $514.236$
Root an. cond. $22.6767$
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  − 2·3-s − 7-s + 9-s − 2·13-s + 2·17-s + 4·19-s + 2·21-s − 23-s + 4·27-s + 6·29-s − 2·31-s + 2·37-s + 4·39-s − 6·41-s − 2·43-s + 12·47-s + 49-s − 4·51-s − 6·53-s − 8·57-s + 8·59-s + 2·61-s − 63-s − 14·67-s + 2·69-s + 12·71-s − 4·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.436·21-s − 0.208·23-s + 0.769·27-s + 1.11·29-s − 0.359·31-s + 0.328·37-s + 0.640·39-s − 0.937·41-s − 0.304·43-s + 1.75·47-s + 1/7·49-s − 0.560·51-s − 0.824·53-s − 1.05·57-s + 1.04·59-s + 0.256·61-s − 0.125·63-s − 1.71·67-s + 0.240·69-s + 1.42·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64400\)    =    \(2^{4} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(514.236\)
Root analytic conductor: \(22.6767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{64400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
23 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.55069089239515, −13.81812089576237, −13.60767852749267, −12.75926469056072, −12.28779770141894, −12.06549118558564, −11.47123176314303, −11.06374140761008, −10.35247735153074, −10.06508828757485, −9.553587486767116, −8.858438578718127, −8.324655300345592, −7.630459265962181, −7.081242428033197, −6.657748786545500, −5.953593534483150, −5.594847399903133, −5.044050450430966, −4.514967875978738, −3.776104842438864, −3.031899896889680, −2.523354254426165, −1.470070041022885, −0.7890420329259930, 0, 0.7890420329259930, 1.470070041022885, 2.523354254426165, 3.031899896889680, 3.776104842438864, 4.514967875978738, 5.044050450430966, 5.594847399903133, 5.953593534483150, 6.657748786545500, 7.081242428033197, 7.630459265962181, 8.324655300345592, 8.858438578718127, 9.553587486767116, 10.06508828757485, 10.35247735153074, 11.06374140761008, 11.47123176314303, 12.06549118558564, 12.28779770141894, 12.75926469056072, 13.60767852749267, 13.81812089576237, 14.55069089239515

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