Properties

Label 2-371280-1.1-c1-0-36
Degree $2$
Conductor $371280$
Sign $1$
Analytic cond. $2964.68$
Root an. cond. $54.4489$
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 + 5-s + 7-s + 9-s + 4·11-s + 13-s − 15-s + 17-s − 4·19-s − 21-s − 8·23-s + 25-s − 27-s − 2·29-s − 4·33-s + 35-s + 6·37-s − 39-s + 10·41-s + 4·43-s + 45-s + 49-s − 51-s − 10·53-s + 4·55-s + 4·57-s + 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.140·51-s − 1.37·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(371280\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(2964.68\)
Root analytic conductor: \(54.4489\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 371280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.813933462\)
\(L(\frac12)\) \(\approx\) \(2.813933462\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
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

−12.42536145394794, −12.02569769069637, −11.64223067919237, −11.08123196133471, −10.78684494490655, −10.32527605727014, −9.751988455861304, −9.261252076313470, −9.104806892082676, −8.349834745617292, −7.771905755524037, −7.647281732731432, −6.744947542183586, −6.332023521022173, −6.112892619298411, −5.649784049090289, −4.978191214446445, −4.468001954646831, −3.973011881428919, −3.698392045918998, −2.787556221245564, −2.120900318430339, −1.752610185417086, −1.066256697358743, −0.4921765066658509, 0.4921765066658509, 1.066256697358743, 1.752610185417086, 2.120900318430339, 2.787556221245564, 3.698392045918998, 3.973011881428919, 4.468001954646831, 4.978191214446445, 5.649784049090289, 6.112892619298411, 6.332023521022173, 6.744947542183586, 7.647281732731432, 7.771905755524037, 8.349834745617292, 9.104806892082676, 9.261252076313470, 9.751988455861304, 10.32527605727014, 10.78684494490655, 11.08123196133471, 11.64223067919237, 12.02569769069637, 12.42536145394794

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