Properties

Label 2-97020-1.1-c1-0-32
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

\(L(1)\) \(\approx\) \(3.639335213\)
\(L(\frac12)\) \(\approx\) \(3.639335213\)
\(L(\frac{3}{2})\) 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

−13.59227836243056, −13.25498158841178, −13.12832900770761, −12.27266222008706, −11.82385129698457, −11.25996193400964, −10.87110000996415, −10.37881672932413, −9.760829152486614, −9.420391727143148, −8.713514498261718, −8.395241368262073, −7.809740918635365, −7.214459824250010, −6.604977513613816, −6.158734503367133, −5.600871883790761, −5.178016646296863, −4.365039020430794, −3.964212167443687, −3.134003921259329, −2.741992855238149, −1.943418979070180, −1.190711568550593, −0.6716529258397893, 0.6716529258397893, 1.190711568550593, 1.943418979070180, 2.741992855238149, 3.134003921259329, 3.964212167443687, 4.365039020430794, 5.178016646296863, 5.600871883790761, 6.158734503367133, 6.604977513613816, 7.214459824250010, 7.809740918635365, 8.395241368262073, 8.713514498261718, 9.420391727143148, 9.760829152486614, 10.37881672932413, 10.87110000996415, 11.25996193400964, 11.82385129698457, 12.27266222008706, 13.12832900770761, 13.25498158841178, 13.59227836243056

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