Properties

Label 2-60984-1.1-c1-0-47
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  + 3·5-s + 7-s − 7·13-s − 2·17-s + 19-s − 6·23-s + 4·25-s + 7·29-s − 10·31-s + 3·35-s − 7·37-s + 6·41-s + 8·43-s + 13·47-s + 49-s + 10·53-s + 9·59-s + 6·61-s − 21·65-s + 67-s − 6·71-s − 13·73-s − 8·79-s + 18·83-s − 6·85-s − 2·89-s − 7·91-s + ⋯
L(s)  = 1  + 1.34·5-s + 0.377·7-s − 1.94·13-s − 0.485·17-s + 0.229·19-s − 1.25·23-s + 4/5·25-s + 1.29·29-s − 1.79·31-s + 0.507·35-s − 1.15·37-s + 0.937·41-s + 1.21·43-s + 1.89·47-s + 1/7·49-s + 1.37·53-s + 1.17·59-s + 0.768·61-s − 2.60·65-s + 0.122·67-s − 0.712·71-s − 1.52·73-s − 0.900·79-s + 1.97·83-s − 0.650·85-s − 0.211·89-s − 0.733·91-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: $\chi_{60984} (1, \cdot )$
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 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−14.47223644058879, −14.07689659262865, −13.67662722445334, −13.03372574966057, −12.53379883201908, −12.03788970518219, −11.69586944134200, −10.76042002185202, −10.36212076550620, −10.06947364143175, −9.301657500093669, −9.152842476214364, −8.428692998138921, −7.690760183950498, −7.181091606863640, −6.823632784333004, −5.910372642360362, −5.565204543515513, −5.145477511407610, −4.367222051559497, −3.911567126006112, −2.768324868394861, −2.311936129183339, −1.976867597294633, −1.013922676095124, 0, 1.013922676095124, 1.976867597294633, 2.311936129183339, 2.768324868394861, 3.911567126006112, 4.367222051559497, 5.145477511407610, 5.565204543515513, 5.910372642360362, 6.823632784333004, 7.181091606863640, 7.690760183950498, 8.428692998138921, 9.152842476214364, 9.301657500093669, 10.06947364143175, 10.36212076550620, 10.76042002185202, 11.69586944134200, 12.03788970518219, 12.53379883201908, 13.03372574966057, 13.67662722445334, 14.07689659262865, 14.47223644058879

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