Properties

Label 2-60984-1.1-c1-0-37
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 − 13-s + 5·17-s + 4·19-s + 4·23-s + 4·25-s − 7·29-s − 4·31-s − 3·35-s − 5·37-s + 9·41-s + 49-s − 3·53-s − 2·61-s + 3·65-s + 8·67-s − 6·73-s − 15·85-s + 13·89-s − 91-s − 12·95-s − 97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s − 0.277·13-s + 1.21·17-s + 0.917·19-s + 0.834·23-s + 4/5·25-s − 1.29·29-s − 0.718·31-s − 0.507·35-s − 0.821·37-s + 1.40·41-s + 1/7·49-s − 0.412·53-s − 0.256·61-s + 0.372·65-s + 0.977·67-s − 0.702·73-s − 1.62·85-s + 1.37·89-s − 0.104·91-s − 1.23·95-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.64781814184186, −14.18998036172329, −13.56198198393469, −12.88143571488664, −12.46081321975824, −11.97612629378204, −11.55429320507054, −11.03885334283115, −10.69023053004304, −9.904101284476199, −9.355028422900854, −8.932728476598125, −8.131546440424899, −7.795609467483752, −7.330947087434467, −7.004567600178700, −6.040061854885849, −5.391688304674241, −5.041709469688070, −4.262324447939460, −3.680133507665726, −3.295107017738471, −2.545318016256335, −1.572536337010777, −0.8843169926989916, 0, 0.8843169926989916, 1.572536337010777, 2.545318016256335, 3.295107017738471, 3.680133507665726, 4.262324447939460, 5.041709469688070, 5.391688304674241, 6.040061854885849, 7.004567600178700, 7.330947087434467, 7.795609467483752, 8.131546440424899, 8.932728476598125, 9.355028422900854, 9.904101284476199, 10.69023053004304, 11.03885334283115, 11.55429320507054, 11.97612629378204, 12.46081321975824, 12.88143571488664, 13.56198198393469, 14.18998036172329, 14.64781814184186

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