Properties

Label 2-62400-1.1-c1-0-103
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 − 3·7-s + 9-s + 3·11-s − 13-s + 3·17-s + 3·21-s − 3·23-s − 27-s − 8·29-s − 4·31-s − 3·33-s − 37-s + 39-s − 3·41-s − 4·43-s + 10·47-s + 2·49-s − 3·51-s + 9·53-s + 4·59-s − 9·61-s − 3·63-s + 4·67-s + 3·69-s − 7·71-s − 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.654·21-s − 0.625·23-s − 0.192·27-s − 1.48·29-s − 0.718·31-s − 0.522·33-s − 0.164·37-s + 0.160·39-s − 0.468·41-s − 0.609·43-s + 1.45·47-s + 2/7·49-s − 0.420·51-s + 1.23·53-s + 0.520·59-s − 1.15·61-s − 0.377·63-s + 0.488·67-s + 0.361·69-s − 0.830·71-s − 0.702·73-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
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

−14.53264679510859, −14.01983304281757, −13.33085874662021, −13.08219456784149, −12.28728062280335, −12.14591781156439, −11.58565099505591, −10.99395302767968, −10.34817496348198, −10.00384884673972, −9.396542113161127, −9.073506507929871, −8.414629743881763, −7.462071950719419, −7.317798636065394, −6.606193688487286, −6.051105867753147, −5.706985226053479, −5.048966063411765, −4.276054553417942, −3.616573564813309, −3.377900810382318, −2.338037758417821, −1.683172146709841, −0.7856529643470219, 0, 0.7856529643470219, 1.683172146709841, 2.338037758417821, 3.377900810382318, 3.616573564813309, 4.276054553417942, 5.048966063411765, 5.706985226053479, 6.051105867753147, 6.606193688487286, 7.317798636065394, 7.462071950719419, 8.414629743881763, 9.073506507929871, 9.396542113161127, 10.00384884673972, 10.34817496348198, 10.99395302767968, 11.58565099505591, 12.14591781156439, 12.28728062280335, 13.08219456784149, 13.33085874662021, 14.01983304281757, 14.53264679510859

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