Properties

Label 2-64400-1.1-c1-0-7
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s − 3·9-s − 2·13-s + 2·17-s − 4·19-s + 23-s − 2·29-s − 2·37-s + 2·41-s + 4·43-s + 12·47-s + 49-s − 2·53-s + 12·59-s − 6·61-s + 3·63-s − 4·67-s + 10·73-s + 4·79-s + 9·81-s − 12·83-s + 6·89-s + 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.208·23-s − 0.371·29-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.274·53-s + 1.56·59-s − 0.768·61-s + 0.377·63-s − 0.488·67-s + 1.17·73-s + 0.450·79-s + 81-s − 1.31·83-s + 0.635·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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: \(0\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.114912752\)
\(L(\frac12)\) \(\approx\) \(1.114912752\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.22907475506321, −13.78969161428472, −13.26187250093678, −12.54538581260806, −12.37785950354832, −11.70525234768979, −11.22851170506125, −10.62368443143439, −10.29181008657156, −9.550665155229039, −9.098696501425060, −8.664132047733152, −8.029675354061516, −7.528657096013062, −6.930575095115748, −6.337942051720846, −5.772741945667074, −5.348424753795458, −4.668102018131566, −3.934894170296631, −3.451755547513577, −2.545652526866848, −2.377972053205796, −1.259490038823917, −0.3659920626756407, 0.3659920626756407, 1.259490038823917, 2.377972053205796, 2.545652526866848, 3.451755547513577, 3.934894170296631, 4.668102018131566, 5.348424753795458, 5.772741945667074, 6.337942051720846, 6.930575095115748, 7.528657096013062, 8.029675354061516, 8.664132047733152, 9.098696501425060, 9.550665155229039, 10.29181008657156, 10.62368443143439, 11.22851170506125, 11.70525234768979, 12.37785950354832, 12.54538581260806, 13.26187250093678, 13.78969161428472, 14.22907475506321

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