Properties

Label 2-60984-1.1-c1-0-15
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  + 7-s + 6·13-s + 2·19-s − 4·23-s − 5·25-s − 2·29-s + 2·31-s + 2·37-s − 8·41-s + 2·47-s + 49-s + 10·53-s + 4·59-s − 10·61-s + 4·67-s + 8·73-s − 8·79-s − 2·83-s + 6·89-s + 6·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.66·13-s + 0.458·19-s − 0.834·23-s − 25-s − 0.371·29-s + 0.359·31-s + 0.328·37-s − 1.24·41-s + 0.291·47-s + 1/7·49-s + 1.37·53-s + 0.520·59-s − 1.28·61-s + 0.488·67-s + 0.936·73-s − 0.900·79-s − 0.219·83-s + 0.635·89-s + 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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

\(L(1)\) \(\approx\) \(2.630556892\)
\(L(\frac12)\) \(\approx\) \(2.630556892\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.16933326538253, −13.70970498996785, −13.45277949494631, −12.87916835340221, −12.15005321871559, −11.70069365916543, −11.34205366571452, −10.76939685850110, −10.18850756980943, −9.803175977130599, −8.999921535823131, −8.669832413985991, −8.038875580262097, −7.682511982187281, −6.952023423237640, −6.311807796484691, −5.866408017377979, −5.359370591222073, −4.633526800990725, −3.852145055824651, −3.666737193879827, −2.769717637086865, −1.952890158366417, −1.402273120304009, −0.5698061886913189, 0.5698061886913189, 1.402273120304009, 1.952890158366417, 2.769717637086865, 3.666737193879827, 3.852145055824651, 4.633526800990725, 5.359370591222073, 5.866408017377979, 6.311807796484691, 6.952023423237640, 7.682511982187281, 8.038875580262097, 8.669832413985991, 8.999921535823131, 9.803175977130599, 10.18850756980943, 10.76939685850110, 11.34205366571452, 11.70069365916543, 12.15005321871559, 12.87916835340221, 13.45277949494631, 13.70970498996785, 14.16933326538253

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