Properties

Label 2-64400-1.1-c1-0-38
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  − 7-s − 3·9-s − 2·11-s − 4·13-s + 6·17-s + 8·19-s + 23-s + 10·29-s − 10·31-s − 8·37-s − 2·41-s + 12·47-s + 49-s + 4·53-s − 14·59-s − 2·61-s + 3·63-s − 4·67-s − 8·71-s + 2·77-s − 6·79-s + 9·81-s − 12·83-s + 10·89-s + 4·91-s + 2·97-s + 6·99-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 0.603·11-s − 1.10·13-s + 1.45·17-s + 1.83·19-s + 0.208·23-s + 1.85·29-s − 1.79·31-s − 1.31·37-s − 0.312·41-s + 1.75·47-s + 1/7·49-s + 0.549·53-s − 1.82·59-s − 0.256·61-s + 0.377·63-s − 0.488·67-s − 0.949·71-s + 0.227·77-s − 0.675·79-s + 81-s − 1.31·83-s + 1.05·89-s + 0.419·91-s + 0.203·97-s + 0.603·99-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 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.46034609882210, −13.92892382745838, −13.72089997366212, −12.90937802236813, −12.31425309366898, −11.91600399824963, −11.79927967898195, −10.71740452446533, −10.50429114880943, −9.931085558955837, −9.324721681816243, −8.990427217640466, −8.248811129395021, −7.665057966470373, −7.327987548437171, −6.799684688279551, −5.827971372325279, −5.517559241895964, −5.153421058330104, −4.423501193786507, −3.444750007552784, −3.048590719456757, −2.678617964323737, −1.685066574153302, −0.8302780139144496, 0, 0.8302780139144496, 1.685066574153302, 2.678617964323737, 3.048590719456757, 3.444750007552784, 4.423501193786507, 5.153421058330104, 5.517559241895964, 5.827971372325279, 6.799684688279551, 7.327987548437171, 7.665057966470373, 8.248811129395021, 8.990427217640466, 9.324721681816243, 9.931085558955837, 10.50429114880943, 10.71740452446533, 11.79927967898195, 11.91600399824963, 12.31425309366898, 12.90937802236813, 13.72089997366212, 13.92892382745838, 14.46034609882210

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