Properties

Label 2-60984-1.1-c1-0-4
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 $0$

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: \(0\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.196282772\)
\(L(\frac12)\) \(\approx\) \(1.196282772\)
\(L(\frac{3}{2})\) 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

−14.22298425415609, −13.81287303785597, −13.15292762535796, −12.82464897458573, −12.29216886351600, −11.71520560769137, −11.11396476941870, −10.89019942891149, −10.52412469324894, −9.443523297878543, −9.172993586874389, −8.466635286126352, −8.183633135466304, −7.633261304657173, −6.813230575298767, −6.584952458913486, −5.980328150056215, −5.066424182716045, −4.681030099821120, −3.883807118538459, −3.355099503715014, −3.211020151093821, −1.978016129180614, −1.247975197097404, −0.4023853670310819, 0.4023853670310819, 1.247975197097404, 1.978016129180614, 3.211020151093821, 3.355099503715014, 3.883807118538459, 4.681030099821120, 5.066424182716045, 5.980328150056215, 6.584952458913486, 6.813230575298767, 7.633261304657173, 8.183633135466304, 8.466635286126352, 9.172993586874389, 9.443523297878543, 10.52412469324894, 10.89019942891149, 11.11396476941870, 11.71520560769137, 12.29216886351600, 12.82464897458573, 13.15292762535796, 13.81287303785597, 14.22298425415609

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