Properties

Label 2-369600-1.1-c1-0-10
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 + 3·13-s − 5·17-s + 19-s + 21-s − 27-s + 6·29-s + 33-s − 5·37-s − 3·39-s + 11·41-s − 8·43-s − 8·47-s + 49-s + 5·51-s − 53-s − 57-s − 10·59-s + 61-s − 63-s − 11·67-s − 9·71-s − 73-s + 77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 1.21·17-s + 0.229·19-s + 0.218·21-s − 0.192·27-s + 1.11·29-s + 0.174·33-s − 0.821·37-s − 0.480·39-s + 1.71·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.700·51-s − 0.137·53-s − 0.132·57-s − 1.30·59-s + 0.128·61-s − 0.125·63-s − 1.34·67-s − 1.06·71-s − 0.117·73-s + 0.113·77-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

\(L(1)\) \(\approx\) \(0.4631867319\)
\(L(\frac12)\) \(\approx\) \(0.4631867319\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−12.54913935781842, −12.02079979640175, −11.51109048056293, −11.20649351353062, −10.67946644709615, −10.36684706364737, −9.847792880658704, −9.358364907347023, −8.799375120603171, −8.510449527978171, −7.926847071652311, −7.360829198619116, −6.878468448106305, −6.438356584289355, −6.047276086901308, −5.616414948719347, −4.926956645144874, −4.521397210982948, −4.117169761572208, −3.351819111721400, −2.968906514893783, −2.322561995232264, −1.572522034765638, −1.133941143982448, −0.1913866603276765, 0.1913866603276765, 1.133941143982448, 1.572522034765638, 2.322561995232264, 2.968906514893783, 3.351819111721400, 4.117169761572208, 4.521397210982948, 4.926956645144874, 5.616414948719347, 6.047276086901308, 6.438356584289355, 6.878468448106305, 7.360829198619116, 7.926847071652311, 8.510449527978171, 8.799375120603171, 9.358364907347023, 9.847792880658704, 10.36684706364737, 10.67946644709615, 11.20649351353062, 11.51109048056293, 12.02079979640175, 12.54913935781842

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