Properties

Label 2-31200-1.1-c1-0-29
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 − 3·11-s + 13-s + 5·17-s − 2·19-s + 21-s + 3·23-s − 27-s − 4·31-s + 3·33-s − 37-s − 39-s + 9·41-s − 2·43-s − 8·47-s − 6·49-s − 5·51-s + 53-s + 2·57-s + 4·59-s − 3·61-s − 63-s + 16·67-s − 3·69-s − 15·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 1.21·17-s − 0.458·19-s + 0.218·21-s + 0.625·23-s − 0.192·27-s − 0.718·31-s + 0.522·33-s − 0.164·37-s − 0.160·39-s + 1.40·41-s − 0.304·43-s − 1.16·47-s − 6/7·49-s − 0.700·51-s + 0.137·53-s + 0.264·57-s + 0.520·59-s − 0.384·61-s − 0.125·63-s + 1.95·67-s − 0.361·69-s − 1.78·71-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 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−15.48442174270927, −14.74537581694553, −14.44164380565018, −13.68309490690509, −13.02724002789915, −12.76893882731391, −12.30480211264893, −11.51964198282411, −11.10648651134826, −10.58444319744109, −9.993911052511972, −9.603051979778604, −8.885296681661176, −8.172756500476899, −7.724120335732944, −7.068720759172142, −6.497822689457012, −5.830117529882027, −5.358454874986873, −4.814484589151862, −3.997960375903337, −3.326568377902020, −2.697884334373437, −1.779300599970711, −0.9142869896112729, 0, 0.9142869896112729, 1.779300599970711, 2.697884334373437, 3.326568377902020, 3.997960375903337, 4.814484589151862, 5.358454874986873, 5.830117529882027, 6.497822689457012, 7.068720759172142, 7.724120335732944, 8.172756500476899, 8.885296681661176, 9.603051979778604, 9.993911052511972, 10.58444319744109, 11.10648651134826, 11.51964198282411, 12.30480211264893, 12.76893882731391, 13.02724002789915, 13.68309490690509, 14.44164380565018, 14.74537581694553, 15.48442174270927

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