Properties

Label 2-31200-1.1-c1-0-50
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 + 9-s + 4·11-s − 13-s − 6·17-s + 4·19-s + 27-s + 2·29-s + 8·31-s + 4·33-s − 6·37-s − 39-s − 10·41-s + 4·43-s − 8·47-s − 7·49-s − 6·51-s − 6·53-s + 4·57-s − 12·59-s − 10·61-s + 4·67-s + 12·71-s + 2·73-s + 8·79-s + 81-s + 8·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.45·17-s + 0.917·19-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.986·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s − 1.16·47-s − 49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s − 1.28·61-s + 0.488·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s + 1/9·81-s + 0.878·83-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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.36246862805395, −14.83570295722025, −14.21262989679450, −13.70801842331846, −13.54935883718087, −12.68009616167130, −12.09944923804040, −11.78299718474599, −11.04905233722285, −10.60093442090079, −9.688902130642022, −9.510453985560249, −8.898800112621104, −8.278975495815761, −7.866644884875084, −6.972638510252509, −6.595556076800141, −6.179166013859373, −4.968808786811848, −4.775321998672803, −3.915257845809373, −3.334288522647703, −2.679370676443683, −1.813017812602212, −1.224095869070800, 0, 1.224095869070800, 1.813017812602212, 2.679370676443683, 3.334288522647703, 3.915257845809373, 4.775321998672803, 4.968808786811848, 6.179166013859373, 6.595556076800141, 6.972638510252509, 7.866644884875084, 8.278975495815761, 8.898800112621104, 9.510453985560249, 9.688902130642022, 10.60093442090079, 11.04905233722285, 11.78299718474599, 12.09944923804040, 12.68009616167130, 13.54935883718087, 13.70801842331846, 14.21262989679450, 14.83570295722025, 15.36246862805395

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