Properties

Label 2-369600-1.1-c1-0-120
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 + 3·17-s − 19-s + 21-s + 7·23-s − 27-s − 29-s + 8·31-s − 33-s − 2·37-s + 2·39-s − 8·41-s + 9·43-s + 12·47-s + 49-s − 3·51-s − 5·53-s + 57-s − 3·59-s − 13·61-s − 63-s − 14·67-s − 7·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.727·17-s − 0.229·19-s + 0.218·21-s + 1.45·23-s − 0.192·27-s − 0.185·29-s + 1.43·31-s − 0.174·33-s − 0.328·37-s + 0.320·39-s − 1.24·41-s + 1.37·43-s + 1.75·47-s + 1/7·49-s − 0.420·51-s − 0.686·53-s + 0.132·57-s − 0.390·59-s − 1.66·61-s − 0.125·63-s − 1.71·67-s − 0.842·69-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

\(L(1)\) \(\approx\) \(1.890209087\)
\(L(\frac12)\) \(\approx\) \(1.890209087\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−12.41155136014429, −12.10198879445375, −11.69962794935114, −11.11235873753346, −10.63427767971846, −10.38779928972993, −9.748665271449533, −9.440217195056833, −8.878648836410330, −8.511203954956780, −7.790302587535517, −7.274538888899203, −7.125808411074961, −6.319950495673876, −6.104870922732637, −5.553220878446843, −4.873696931513695, −4.685776294306474, −4.019345388543457, −3.384484232629350, −2.918158488714172, −2.390312741331394, −1.566321052964773, −1.022777284068887, −0.4236149439407358, 0.4236149439407358, 1.022777284068887, 1.566321052964773, 2.390312741331394, 2.918158488714172, 3.384484232629350, 4.019345388543457, 4.685776294306474, 4.873696931513695, 5.553220878446843, 6.104870922732637, 6.319950495673876, 7.125808411074961, 7.274538888899203, 7.790302587535517, 8.511203954956780, 8.878648836410330, 9.440217195056833, 9.748665271449533, 10.38779928972993, 10.63427767971846, 11.11235873753346, 11.69962794935114, 12.10198879445375, 12.41155136014429

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