Properties

Label 2-369600-1.1-c1-0-117
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 11-s + 13-s + 5·17-s − 8·19-s − 21-s + 23-s − 27-s − 3·29-s − 3·31-s + 33-s + 4·37-s − 39-s + 3·41-s + 9·43-s − 2·47-s + 49-s − 5·51-s + 9·53-s + 8·57-s + 5·59-s + 5·61-s + 63-s − 12·67-s − 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 1.21·17-s − 1.83·19-s − 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.557·29-s − 0.538·31-s + 0.174·33-s + 0.657·37-s − 0.160·39-s + 0.468·41-s + 1.37·43-s − 0.291·47-s + 1/7·49-s − 0.700·51-s + 1.23·53-s + 1.05·57-s + 0.650·59-s + 0.640·61-s + 0.125·63-s − 1.46·67-s − 0.120·69-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: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.939996815\)
\(L(\frac12)\) \(\approx\) \(1.939996815\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−12.52829253290903, −12.07407553664472, −11.56413263712854, −11.07850600655952, −10.75091328492817, −10.35883451615268, −9.909772213340027, −9.317276155927289, −8.852034340729030, −8.394967027184234, −7.886598829723193, −7.413296411234435, −7.062492253120945, −6.334295773552220, −5.966860792017754, −5.563058530181281, −5.090691443705448, −4.411406587172581, −4.096013005461359, −3.558412316698947, −2.807274006065120, −2.263971707051745, −1.696823453795201, −1.012521849966361, −0.4215443975640144, 0.4215443975640144, 1.012521849966361, 1.696823453795201, 2.263971707051745, 2.807274006065120, 3.558412316698947, 4.096013005461359, 4.411406587172581, 5.090691443705448, 5.563058530181281, 5.966860792017754, 6.334295773552220, 7.062492253120945, 7.413296411234435, 7.886598829723193, 8.394967027184234, 8.852034340729030, 9.317276155927289, 9.909772213340027, 10.35883451615268, 10.75091328492817, 11.07850600655952, 11.56413263712854, 12.07407553664472, 12.52829253290903

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