Properties

Label 2-369600-1.1-c1-0-121
Degree $2$
Conductor $369600$
Sign $-1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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 − 7-s + 9-s − 11-s − 4·13-s − 4·17-s − 4·19-s + 21-s − 2·23-s − 27-s − 10·29-s − 4·31-s + 33-s + 4·37-s + 4·39-s − 6·41-s + 49-s + 4·51-s − 6·53-s + 4·57-s − 12·59-s + 14·61-s − 63-s + 2·67-s + 2·69-s + 12·71-s − 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.970·17-s − 0.917·19-s + 0.218·21-s − 0.417·23-s − 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.174·33-s + 0.657·37-s + 0.640·39-s − 0.937·41-s + 1/7·49-s + 0.560·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s + 1.79·61-s − 0.125·63-s + 0.244·67-s + 0.240·69-s + 1.42·71-s − 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−12.66605254003604, −12.36582609751932, −11.81841605242210, −11.29537886442156, −10.95188359834005, −10.58928629766992, −9.980143181274653, −9.550295038745400, −9.293618596023836, −8.621406990718498, −8.157546523509262, −7.526267382907660, −7.262137693118459, −6.619982843795109, −6.336936032024227, −5.737983083601686, −5.197445240809387, −4.891862137568460, −4.146147331481811, −3.901789828523036, −3.161853137111387, −2.457042114291156, −2.057907111246929, −1.514325464867429, −0.4241539145650760, 0, 0.4241539145650760, 1.514325464867429, 2.057907111246929, 2.457042114291156, 3.161853137111387, 3.901789828523036, 4.146147331481811, 4.891862137568460, 5.197445240809387, 5.737983083601686, 6.336936032024227, 6.619982843795109, 7.262137693118459, 7.526267382907660, 8.157546523509262, 8.621406990718498, 9.293618596023836, 9.550295038745400, 9.980143181274653, 10.58928629766992, 10.95188359834005, 11.29537886442156, 11.81841605242210, 12.36582609751932, 12.66605254003604

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