Properties

Label 2-97020-1.1-c1-0-2
Degree $2$
Conductor $97020$
Sign $1$
Analytic cond. $774.708$
Root an. cond. $27.8335$
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  − 5-s − 11-s − 5·13-s − 2·19-s + 3·23-s + 25-s − 3·29-s − 8·31-s + 8·37-s + 3·41-s + 5·43-s + 3·47-s + 9·53-s + 55-s − 2·61-s + 5·65-s + 14·67-s − 12·71-s + 10·73-s + 8·79-s + 6·83-s + 2·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 1.38·13-s − 0.458·19-s + 0.625·23-s + 1/5·25-s − 0.557·29-s − 1.43·31-s + 1.31·37-s + 0.468·41-s + 0.762·43-s + 0.437·47-s + 1.23·53-s + 0.134·55-s − 0.256·61-s + 0.620·65-s + 1.71·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s + 0.658·83-s + 0.205·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97020\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(774.708\)
Root analytic conductor: \(27.8335\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{97020} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188887477\)
\(L(\frac12)\) \(\approx\) \(1.188887477\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.81744555835850, −13.11180549082658, −12.83044553183120, −12.35704180960572, −11.88526557306695, −11.29243337711090, −10.83495870740154, −10.47657181549774, −9.699932931967260, −9.366790512224141, −8.898227438273448, −8.194639484710956, −7.641772613352102, −7.369422063261582, −6.803826723476872, −6.178721735280787, −5.381422354758906, −5.191635974075055, −4.360034943654602, −3.985406380889820, −3.277044877032449, −2.450472274273692, −2.255350633551797, −1.163168309495138, −0.3654099675146742, 0.3654099675146742, 1.163168309495138, 2.255350633551797, 2.450472274273692, 3.277044877032449, 3.985406380889820, 4.360034943654602, 5.191635974075055, 5.381422354758906, 6.178721735280787, 6.803826723476872, 7.369422063261582, 7.641772613352102, 8.194639484710956, 8.898227438273448, 9.366790512224141, 9.699932931967260, 10.47657181549774, 10.83495870740154, 11.29243337711090, 11.88526557306695, 12.35704180960572, 12.83044553183120, 13.11180549082658, 13.81744555835850

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