Properties

Label 2-62400-1.1-c1-0-69
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 − 4·7-s + 9-s − 2·11-s + 13-s − 4·17-s − 2·19-s + 4·21-s + 6·23-s − 27-s + 2·29-s − 4·31-s + 2·33-s − 6·37-s − 39-s − 6·41-s + 8·43-s − 8·47-s + 9·49-s + 4·51-s + 10·53-s + 2·57-s + 14·59-s − 10·61-s − 4·63-s + 4·67-s − 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.970·17-s − 0.458·19-s + 0.872·21-s + 1.25·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.348·33-s − 0.986·37-s − 0.160·39-s − 0.937·41-s + 1.21·43-s − 1.16·47-s + 9/7·49-s + 0.560·51-s + 1.37·53-s + 0.264·57-s + 1.82·59-s − 1.28·61-s − 0.503·63-s + 0.488·67-s − 0.722·69-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.65775640458320, −13.74528134836115, −13.41165763370987, −12.95289800149272, −12.62655353219589, −12.11458091527693, −11.37564169446532, −10.97467896540117, −10.44737351767235, −10.02033792910855, −9.480605412890657, −8.787456997101811, −8.601187029844751, −7.634992663164785, −6.961094954973343, −6.750076868389122, −6.203367509739489, −5.501103686522403, −5.122542770628501, −4.300351215014197, −3.758473607663157, −3.067080286277707, −2.538398752756803, −1.694174365995474, −0.6691563813121786, 0, 0.6691563813121786, 1.694174365995474, 2.538398752756803, 3.067080286277707, 3.758473607663157, 4.300351215014197, 5.122542770628501, 5.501103686522403, 6.203367509739489, 6.750076868389122, 6.961094954973343, 7.634992663164785, 8.601187029844751, 8.787456997101811, 9.480605412890657, 10.02033792910855, 10.44737351767235, 10.97467896540117, 11.37564169446532, 12.11458091527693, 12.62655353219589, 12.95289800149272, 13.41165763370987, 13.74528134836115, 14.65775640458320

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