Properties

Label 2-31200-1.1-c1-0-21
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 + 5·7-s + 9-s − 11-s − 13-s − 3·17-s + 6·19-s − 5·21-s + 3·23-s − 27-s + 4·29-s + 33-s + 5·37-s + 39-s + 11·41-s + 6·43-s + 18·49-s + 3·51-s + 9·53-s − 6·57-s − 12·59-s + 5·61-s + 5·63-s + 8·67-s − 3·69-s − 13·71-s + 10·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.88·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.727·17-s + 1.37·19-s − 1.09·21-s + 0.625·23-s − 0.192·27-s + 0.742·29-s + 0.174·33-s + 0.821·37-s + 0.160·39-s + 1.71·41-s + 0.914·43-s + 18/7·49-s + 0.420·51-s + 1.23·53-s − 0.794·57-s − 1.56·59-s + 0.640·61-s + 0.629·63-s + 0.977·67-s − 0.361·69-s − 1.54·71-s + 1.17·73-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

\(L(1)\) \(\approx\) \(2.910728352\)
\(L(\frac12)\) \(\approx\) \(2.910728352\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
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

−15.13794578288921, −14.50269639436867, −14.09881344586698, −13.60291058632641, −12.94143936888654, −12.29387067160111, −11.78965929938363, −11.38389879818224, −10.77578997256303, −10.64332783986716, −9.630722924241096, −9.209729202731900, −8.486871167604373, −7.878971196534319, −7.498158986136399, −6.947258863061363, −6.082655245986262, −5.462036750198485, −5.003517087462467, −4.498876327813326, −3.935779529560526, −2.769725764594552, −2.242079171376778, −1.273371430108238, −0.7697808345624478, 0.7697808345624478, 1.273371430108238, 2.242079171376778, 2.769725764594552, 3.935779529560526, 4.498876327813326, 5.003517087462467, 5.462036750198485, 6.082655245986262, 6.947258863061363, 7.498158986136399, 7.878971196534319, 8.486871167604373, 9.209729202731900, 9.630722924241096, 10.64332783986716, 10.77578997256303, 11.38389879818224, 11.78965929938363, 12.29387067160111, 12.94143936888654, 13.60291058632641, 14.09881344586698, 14.50269639436867, 15.13794578288921

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