Properties

Label 2-60984-1.1-c1-0-41
Degree $2$
Conductor $60984$
Sign $-1$
Analytic cond. $486.959$
Root an. cond. $22.0671$
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  + 5-s − 7-s + 5·13-s − 6·17-s + 19-s − 4·23-s − 4·25-s + 29-s − 10·31-s − 35-s + 37-s + 8·43-s − 47-s + 49-s − 8·53-s + 3·59-s + 6·61-s + 5·65-s + 13·67-s + 3·73-s + 8·79-s + 2·83-s − 6·85-s + 6·89-s − 5·91-s + 95-s − 16·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.38·13-s − 1.45·17-s + 0.229·19-s − 0.834·23-s − 4/5·25-s + 0.185·29-s − 1.79·31-s − 0.169·35-s + 0.164·37-s + 1.21·43-s − 0.145·47-s + 1/7·49-s − 1.09·53-s + 0.390·59-s + 0.768·61-s + 0.620·65-s + 1.58·67-s + 0.351·73-s + 0.900·79-s + 0.219·83-s − 0.650·85-s + 0.635·89-s − 0.524·91-s + 0.102·95-s − 1.62·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60984\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(486.959\)
Root analytic conductor: \(22.0671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 - T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−14.34671098584056, −14.04137232802669, −13.49279870422066, −13.06979905353787, −12.71312445759362, −12.02531726426404, −11.37981152082640, −10.93285458130350, −10.67258448119149, −9.790214295320350, −9.451997101841992, −8.965012644728895, −8.374643849377530, −7.901560077833381, −7.148455508178662, −6.615622965559943, −6.101221148156102, −5.696750317513277, −5.046111266766255, −4.156528247210862, −3.838027288785030, −3.179240306727282, −2.202133541030622, −1.912596261004074, −0.9288242019688921, 0, 0.9288242019688921, 1.912596261004074, 2.202133541030622, 3.179240306727282, 3.838027288785030, 4.156528247210862, 5.046111266766255, 5.696750317513277, 6.101221148156102, 6.615622965559943, 7.148455508178662, 7.901560077833381, 8.374643849377530, 8.965012644728895, 9.451997101841992, 9.790214295320350, 10.67258448119149, 10.93285458130350, 11.37981152082640, 12.02531726426404, 12.71312445759362, 13.06979905353787, 13.49279870422066, 14.04137232802669, 14.34671098584056

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