Properties

Degree $2$
Conductor $346560$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 2·7-s + 9-s − 2·13-s + 15-s − 6·17-s − 2·21-s − 6·23-s + 25-s + 27-s + 4·29-s − 2·35-s − 10·37-s − 2·39-s − 8·41-s + 2·43-s + 45-s + 2·47-s − 3·49-s − 6·51-s + 2·53-s − 14·61-s − 2·63-s − 2·65-s + 4·67-s − 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.436·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.338·35-s − 1.64·37-s − 0.320·39-s − 1.24·41-s + 0.304·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s − 0.840·51-s + 0.274·53-s − 1.79·61-s − 0.251·63-s − 0.248·65-s + 0.488·67-s − 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(346560\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{346560} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 346560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3986694814\)
\(L(\frac12)\) \(\approx\) \(0.3986694814\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−12.49218705604244, −12.23149272142705, −11.80519131514449, −11.12491777692652, −10.51496845793870, −10.35351359812800, −9.720002223593001, −9.428881235594453, −8.961706502707792, −8.349598966973214, −8.242282800928397, −7.293979076098523, −7.086746076535551, −6.506348706182292, −6.164045387674838, −5.593655231785976, −4.873180405719020, −4.570225213902294, −3.919774679006808, −3.379042399323721, −2.909557910139222, −2.241300461890812, −1.951955070238437, −1.232453499924457, −0.1494866429404325, 0.1494866429404325, 1.232453499924457, 1.951955070238437, 2.241300461890812, 2.909557910139222, 3.379042399323721, 3.919774679006808, 4.570225213902294, 4.873180405719020, 5.593655231785976, 6.164045387674838, 6.506348706182292, 7.086746076535551, 7.293979076098523, 8.242282800928397, 8.349598966973214, 8.961706502707792, 9.428881235594453, 9.720002223593001, 10.35351359812800, 10.51496845793870, 11.12491777692652, 11.80519131514449, 12.23149272142705, 12.49218705604244

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