Properties

Label 2-97020-1.1-c1-0-8
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 − 6·13-s − 6·17-s + 8·23-s + 25-s + 2·29-s + 8·31-s + 6·37-s + 10·41-s + 6·53-s − 55-s − 2·59-s − 8·61-s + 6·65-s − 2·67-s + 8·71-s − 14·73-s − 10·79-s − 6·83-s + 6·85-s + 6·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.301·11-s − 1.66·13-s − 1.45·17-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.986·37-s + 1.56·41-s + 0.824·53-s − 0.134·55-s − 0.260·59-s − 1.02·61-s + 0.744·65-s − 0.244·67-s + 0.949·71-s − 1.63·73-s − 1.12·79-s − 0.658·83-s + 0.650·85-s + 0.635·89-s − 0.812·97-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

\(L(1)\) \(\approx\) \(1.647148731\)
\(L(\frac12)\) \(\approx\) \(1.647148731\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.76239982828445, −13.23294698886942, −12.81011043407709, −12.30035571398322, −11.83341827171877, −11.36767758099672, −10.87354246890199, −10.42943453985901, −9.678476599917598, −9.399857141698261, −8.796688560975871, −8.359641953881646, −7.638705168542017, −7.238880153246200, −6.793079343605710, −6.262746445942823, −5.553558691164469, −4.859716499954691, −4.408270702768878, −4.187619676679642, −2.926630551033884, −2.833914888037208, −2.108783794486256, −1.136159288957201, −0.4426847337850786, 0.4426847337850786, 1.136159288957201, 2.108783794486256, 2.833914888037208, 2.926630551033884, 4.187619676679642, 4.408270702768878, 4.859716499954691, 5.553558691164469, 6.262746445942823, 6.793079343605710, 7.238880153246200, 7.638705168542017, 8.359641953881646, 8.796688560975871, 9.399857141698261, 9.678476599917598, 10.42943453985901, 10.87354246890199, 11.36767758099672, 11.83341827171877, 12.30035571398322, 12.81011043407709, 13.23294698886942, 13.76239982828445

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