Properties

Label 2-31200-1.1-c1-0-33
Degree $2$
Conductor $31200$
Sign $-1$
Analytic cond. $249.133$
Root an. cond. $15.7839$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s − 5·11-s − 13-s − 3·17-s − 21-s − 4·23-s + 27-s + 9·29-s + 7·31-s − 5·33-s + 8·37-s − 39-s − 2·41-s + 7·47-s − 6·49-s − 3·51-s + 3·53-s + 9·59-s − 15·61-s − 63-s − 7·67-s − 4·69-s + 4·73-s + 5·77-s + 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s − 0.727·17-s − 0.218·21-s − 0.834·23-s + 0.192·27-s + 1.67·29-s + 1.25·31-s − 0.870·33-s + 1.31·37-s − 0.160·39-s − 0.312·41-s + 1.02·47-s − 6/7·49-s − 0.420·51-s + 0.412·53-s + 1.17·59-s − 1.92·61-s − 0.125·63-s − 0.855·67-s − 0.481·69-s + 0.468·73-s + 0.569·77-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31200\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(249.133\)
Root analytic conductor: \(15.7839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{31200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.43784157423798, −14.93032927881182, −14.20155510427840, −13.63535404771922, −13.42049707325886, −12.77156552625102, −12.24637325035401, −11.75920196045428, −10.94239840352224, −10.39703718226162, −10.05713106622401, −9.465668426619100, −8.816959459903746, −8.104546844967849, −7.954350289043437, −7.191132620137920, −6.494643016232367, −6.039637130152059, −5.181802351264483, −4.602771027383627, −4.090965686880208, −3.081078954787288, −2.663573848917819, −2.137844202146896, −0.9808326031737045, 0, 0.9808326031737045, 2.137844202146896, 2.663573848917819, 3.081078954787288, 4.090965686880208, 4.602771027383627, 5.181802351264483, 6.039637130152059, 6.494643016232367, 7.191132620137920, 7.954350289043437, 8.104546844967849, 8.816959459903746, 9.465668426619100, 10.05713106622401, 10.39703718226162, 10.94239840352224, 11.75920196045428, 12.24637325035401, 12.77156552625102, 13.42049707325886, 13.63535404771922, 14.20155510427840, 14.93032927881182, 15.43784157423798

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