Properties

Label 2-62400-1.1-c1-0-201
Degree $2$
Conductor $62400$
Sign $1$
Analytic cond. $498.266$
Root an. cond. $22.3218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s + 13-s − 2·17-s − 8·19-s + 4·21-s − 8·23-s − 27-s + 2·29-s + 4·31-s − 10·37-s − 39-s + 2·41-s − 4·43-s + 12·47-s + 9·49-s + 2·51-s + 6·53-s + 8·57-s + 2·61-s − 4·63-s + 8·67-s + 8·69-s − 12·71-s − 10·73-s − 8·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.485·17-s − 1.83·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 1.64·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 1.05·57-s + 0.256·61-s − 0.503·63-s + 0.977·67-s + 0.963·69-s − 1.42·71-s − 1.17·73-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62400\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(498.266\)
Root analytic conductor: \(22.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{62400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ 1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.86457897823593, −14.15277193812616, −13.64048629410057, −13.23010837887524, −12.69526536888848, −12.26763666089567, −11.91288484582904, −11.21035459879078, −10.54122528526640, −10.21344341681205, −9.926451265491385, −9.075117548951101, −8.667011179541290, −8.193511630072473, −7.287523731487965, −6.815440258382812, −6.369150953571118, −5.948266613126758, −5.454213094267901, −4.469377647802336, −4.083599435780209, −3.569667861626527, −2.662606767037531, −2.183597634812718, −1.243373390138706, 0, 0, 1.243373390138706, 2.183597634812718, 2.662606767037531, 3.569667861626527, 4.083599435780209, 4.469377647802336, 5.454213094267901, 5.948266613126758, 6.369150953571118, 6.815440258382812, 7.287523731487965, 8.193511630072473, 8.667011179541290, 9.075117548951101, 9.926451265491385, 10.21344341681205, 10.54122528526640, 11.21035459879078, 11.91288484582904, 12.26763666089567, 12.69526536888848, 13.23010837887524, 13.64048629410057, 14.15277193812616, 14.86457897823593

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