Properties

Label 2-346560-1.1-c1-0-262
Degree $2$
Conductor $346560$
Sign $-1$
Analytic cond. $2767.29$
Root an. cond. $52.6050$
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 + 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: $-1$
Analytic conductor: \(2767.29\)
Root analytic conductor: \(52.6050\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{346560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 346560,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
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

−12.91247559473192, −12.53807184524025, −11.73403232227459, −11.27693877524548, −11.03143518684159, −10.64887255102056, −10.02272604593230, −9.419089750552589, −9.054128578478484, −8.869772543472594, −8.172391412247636, −7.782241976534051, −7.352410691986455, −6.771182621314337, −6.200047835407550, −5.967418677686506, −5.078689786545865, −4.781256491819408, −4.306395006000441, −3.731665578743974, −3.105522140854550, −2.514217254651073, −2.127542152504985, −1.428674579661817, −1.001499736895136, 0, 1.001499736895136, 1.428674579661817, 2.127542152504985, 2.514217254651073, 3.105522140854550, 3.731665578743974, 4.306395006000441, 4.781256491819408, 5.078689786545865, 5.967418677686506, 6.200047835407550, 6.771182621314337, 7.352410691986455, 7.782241976534051, 8.172391412247636, 8.869772543472594, 9.054128578478484, 9.419089750552589, 10.02272604593230, 10.64887255102056, 11.03143518684159, 11.27693877524548, 11.73403232227459, 12.53807184524025, 12.91247559473192

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