Properties

Label 2-62400-1.1-c1-0-133
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 13-s − 2·17-s + 4·19-s − 4·23-s − 27-s + 6·29-s + 8·31-s − 6·37-s + 39-s − 2·41-s + 4·43-s − 7·49-s + 2·51-s + 6·53-s − 4·57-s + 2·61-s + 8·67-s + 4·69-s − 6·73-s − 4·79-s + 81-s − 12·83-s − 6·87-s + 6·89-s − 8·93-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.986·37-s + 0.160·39-s − 0.312·41-s + 0.609·43-s − 49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s + 0.256·61-s + 0.977·67-s + 0.481·69-s − 0.702·73-s − 0.450·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + 0.635·89-s − 0.829·93-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: \(1\)
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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.40916747577799, −14.00000073638801, −13.52953258890552, −13.02970667446970, −12.31766825823115, −11.97488750033931, −11.63633319437021, −10.97372414867322, −10.44823493975412, −9.914595335948272, −9.639269750725598, −8.794521157025954, −8.341897644687547, −7.786091141044085, −7.131522514869877, −6.640927748836281, −6.176257603493072, −5.470186894141288, −5.019554846236469, −4.400267155511714, −3.862880417332507, −3.019597982607607, −2.471291143436893, −1.597072996677803, −0.8888460189134461, 0, 0.8888460189134461, 1.597072996677803, 2.471291143436893, 3.019597982607607, 3.862880417332507, 4.400267155511714, 5.019554846236469, 5.470186894141288, 6.176257603493072, 6.640927748836281, 7.131522514869877, 7.786091141044085, 8.341897644687547, 8.794521157025954, 9.639269750725598, 9.914595335948272, 10.44823493975412, 10.97372414867322, 11.63633319437021, 11.97488750033931, 12.31766825823115, 13.02970667446970, 13.52953258890552, 14.00000073638801, 14.40916747577799

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