Properties

Degree $2$
Conductor $3920$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 3·9-s − 4·11-s + 6·13-s − 2·17-s + 25-s + 6·29-s + 8·31-s − 10·37-s − 2·41-s − 4·43-s − 3·45-s + 8·47-s − 2·53-s − 4·55-s − 8·59-s + 14·61-s + 6·65-s + 12·67-s + 16·71-s − 2·73-s + 8·79-s + 9·81-s + 8·83-s − 2·85-s − 10·89-s − 2·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.16·47-s − 0.274·53-s − 0.539·55-s − 1.04·59-s + 1.79·61-s + 0.744·65-s + 1.46·67-s + 1.89·71-s − 0.234·73-s + 0.900·79-s + 81-s + 0.878·83-s − 0.216·85-s − 1.05·89-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3920} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.784452441\)
\(L(\frac12)\) \(\approx\) \(1.784452441\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−18.12659075934542, −17.53823860923922, −17.08312786859915, −16.21594382253087, −15.57209094712168, −15.38005252726752, −14.07366678614862, −13.90751573858575, −13.31256982346756, −12.57905450698390, −11.81010812402662, −11.08364218810912, −10.59209356642135, −9.986591123443433, −8.985245806073316, −8.400283535029584, −8.058922554914412, −6.782480700301191, −6.281362928176271, −5.472549391857356, −4.900546866171145, −3.721799237731026, −2.922578498086050, −2.096637600258421, −0.7596095834913860, 0.7596095834913860, 2.096637600258421, 2.922578498086050, 3.721799237731026, 4.900546866171145, 5.472549391857356, 6.281362928176271, 6.782480700301191, 8.058922554914412, 8.400283535029584, 8.985245806073316, 9.986591123443433, 10.59209356642135, 11.08364218810912, 11.81010812402662, 12.57905450698390, 13.31256982346756, 13.90751573858575, 14.07366678614862, 15.38005252726752, 15.57209094712168, 16.21594382253087, 17.08312786859915, 17.53823860923922, 18.12659075934542

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