Properties

Label 2-369600-1.1-c1-0-101
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 − 2·13-s + 6·17-s − 8·19-s + 21-s + 6·23-s + 27-s + 6·29-s + 10·31-s − 33-s − 6·37-s − 2·39-s − 8·43-s − 8·47-s + 49-s + 6·51-s − 10·53-s − 8·57-s − 10·59-s − 4·61-s + 63-s + 14·67-s + 6·69-s − 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 1.45·17-s − 1.83·19-s + 0.218·21-s + 1.25·23-s + 0.192·27-s + 1.11·29-s + 1.79·31-s − 0.174·33-s − 0.986·37-s − 0.320·39-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.840·51-s − 1.37·53-s − 1.05·57-s − 1.30·59-s − 0.512·61-s + 0.125·63-s + 1.71·67-s + 0.722·69-s − 1.42·71-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

\(L(1)\) \(\approx\) \(2.584969637\)
\(L(\frac12)\) \(\approx\) \(2.584969637\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.41205003167461, −12.16124523355202, −11.73194713208284, −11.02534110000049, −10.61067846641257, −10.26647427533729, −9.792830671659654, −9.389215503544563, −8.673862713554362, −8.430492593859514, −7.900525869495961, −7.725700196125302, −6.843508712428952, −6.595534621674487, −6.165482509710154, −5.264706218970841, −4.983496303269632, −4.554322536389609, −3.997626698207358, −3.231013512426237, −2.928667924983342, −2.444216429775474, −1.617230591099643, −1.304804142376591, −0.3906355228236288, 0.3906355228236288, 1.304804142376591, 1.617230591099643, 2.444216429775474, 2.928667924983342, 3.231013512426237, 3.997626698207358, 4.554322536389609, 4.983496303269632, 5.264706218970841, 6.165482509710154, 6.595534621674487, 6.843508712428952, 7.725700196125302, 7.900525869495961, 8.430492593859514, 8.673862713554362, 9.389215503544563, 9.792830671659654, 10.26647427533729, 10.61067846641257, 11.02534110000049, 11.73194713208284, 12.16124523355202, 12.41205003167461

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