Properties

Label 2-64400-1.1-c1-0-51
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 + 6·11-s + 13-s + 3·17-s + 4·19-s − 2·21-s − 23-s + 4·27-s − 6·29-s − 8·31-s − 12·33-s − 11·37-s − 2·39-s + 8·43-s + 3·47-s + 49-s − 6·51-s − 9·53-s − 8·57-s + 8·61-s + 63-s + 8·67-s + 2·69-s + 6·71-s − 2·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s + 0.277·13-s + 0.727·17-s + 0.917·19-s − 0.436·21-s − 0.208·23-s + 0.769·27-s − 1.11·29-s − 1.43·31-s − 2.08·33-s − 1.80·37-s − 0.320·39-s + 1.21·43-s + 0.437·47-s + 1/7·49-s − 0.840·51-s − 1.23·53-s − 1.05·57-s + 1.02·61-s + 0.125·63-s + 0.977·67-s + 0.240·69-s + 0.712·71-s − 0.234·73-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 \( 1 - 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−14.46047314962433, −14.09477238050588, −13.58842293579607, −12.66259374428982, −12.37811002940882, −11.94526041000197, −11.36752164276065, −11.13738521748287, −10.66116940437071, −9.818037611799621, −9.511838927999762, −8.839333899927818, −8.462083187558225, −7.479127533192119, −7.235165724344436, −6.556962660921259, −6.037399241726173, −5.489625808462029, −5.181983636922830, −4.373638447829154, −3.656424633257555, −3.432064230788627, −2.210511991261009, −1.435594414866192, −0.9998239250530867, 0, 0.9998239250530867, 1.435594414866192, 2.210511991261009, 3.432064230788627, 3.656424633257555, 4.373638447829154, 5.181983636922830, 5.489625808462029, 6.037399241726173, 6.556962660921259, 7.235165724344436, 7.479127533192119, 8.462083187558225, 8.839333899927818, 9.511838927999762, 9.818037611799621, 10.66116940437071, 11.13738521748287, 11.36752164276065, 11.94526041000197, 12.37811002940882, 12.66259374428982, 13.58842293579607, 14.09477238050588, 14.46047314962433

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