Properties

Label 2-31200-1.1-c1-0-23
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 − 2·7-s + 9-s − 2·11-s + 13-s − 6·17-s + 2·19-s + 2·21-s − 27-s + 2·29-s − 2·31-s + 2·33-s − 10·37-s − 39-s + 2·41-s + 8·43-s − 2·47-s − 3·49-s + 6·51-s + 2·53-s − 2·57-s − 10·59-s + 6·61-s − 2·63-s + 14·67-s + 14·71-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 0.192·27-s + 0.371·29-s − 0.359·31-s + 0.348·33-s − 1.64·37-s − 0.160·39-s + 0.312·41-s + 1.21·43-s − 0.291·47-s − 3/7·49-s + 0.840·51-s + 0.274·53-s − 0.264·57-s − 1.30·59-s + 0.768·61-s − 0.251·63-s + 1.71·67-s + 1.66·71-s + 0.702·73-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−15.58606537474117, −15.01029480001453, −14.06701209152272, −13.81116352266592, −13.12220739684228, −12.70043368352472, −12.30033541218368, −11.56603097016458, −11.00669813196282, −10.66496909854000, −10.05634034568496, −9.386563551642428, −9.024326190578648, −8.244830990826020, −7.696638904793619, −6.834642729130401, −6.653558992532730, −5.952927809812633, −5.252880397254994, −4.821143052678029, −3.957314961489214, −3.433282148373101, −2.557960923681036, −1.924748860886868, −0.8122358197898018, 0, 0.8122358197898018, 1.924748860886868, 2.557960923681036, 3.433282148373101, 3.957314961489214, 4.821143052678029, 5.252880397254994, 5.952927809812633, 6.653558992532730, 6.834642729130401, 7.696638904793619, 8.244830990826020, 9.024326190578648, 9.386563551642428, 10.05634034568496, 10.66496909854000, 11.00669813196282, 11.56603097016458, 12.30033541218368, 12.70043368352472, 13.12220739684228, 13.81116352266592, 14.06701209152272, 15.01029480001453, 15.58606537474117

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