Properties

Label 2-31200-1.1-c1-0-11
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 3·11-s + 13-s + 3·17-s + 6·19-s − 21-s − 23-s − 27-s + 8·29-s − 4·31-s + 3·33-s − 5·37-s − 39-s − 5·41-s + 6·43-s − 8·47-s − 6·49-s − 3·51-s − 9·53-s − 6·57-s − 4·59-s − 11·61-s + 63-s + 16·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.727·17-s + 1.37·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.522·33-s − 0.821·37-s − 0.160·39-s − 0.780·41-s + 0.914·43-s − 1.16·47-s − 6/7·49-s − 0.420·51-s − 1.23·53-s − 0.794·57-s − 0.520·59-s − 1.40·61-s + 0.125·63-s + 1.95·67-s + 0.120·69-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: \(0\)
Selberg data: \((2,\ 31200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.778703103\)
\(L(\frac12)\) \(\approx\) \(1.778703103\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−15.21705865292568, −14.47111679614140, −13.93685397626403, −13.67667406877796, −12.78883565851569, −12.45438840668436, −11.90896507771912, −11.34362138906868, −10.83124543706488, −10.36279852759450, −9.721775832626943, −9.340612353279227, −8.376348238397082, −7.969808475496469, −7.505432525573932, −6.754733809061990, −6.211210532085976, −5.466905877098149, −5.040585934348575, −4.613099551722496, −3.476780899146587, −3.187086353577058, −2.137335713671756, −1.369756269075145, −0.5577869994922256, 0.5577869994922256, 1.369756269075145, 2.137335713671756, 3.187086353577058, 3.476780899146587, 4.613099551722496, 5.040585934348575, 5.466905877098149, 6.211210532085976, 6.754733809061990, 7.505432525573932, 7.969808475496469, 8.376348238397082, 9.340612353279227, 9.721775832626943, 10.36279852759450, 10.83124543706488, 11.34362138906868, 11.90896507771912, 12.45438840668436, 12.78883565851569, 13.67667406877796, 13.93685397626403, 14.47111679614140, 15.21705865292568

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