Properties

Label 2-97020-1.1-c1-0-43
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 11-s − 6·13-s + 5·17-s + 19-s + 23-s + 25-s − 3·29-s − 2·37-s + 2·41-s + 43-s + 6·47-s + 53-s + 55-s − 9·59-s − 13·61-s + 6·65-s + 8·67-s − 2·71-s − 10·73-s + 16·79-s − 3·83-s − 5·85-s + 89-s − 95-s + 97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s − 1.66·13-s + 1.21·17-s + 0.229·19-s + 0.208·23-s + 1/5·25-s − 0.557·29-s − 0.328·37-s + 0.312·41-s + 0.152·43-s + 0.875·47-s + 0.137·53-s + 0.134·55-s − 1.17·59-s − 1.66·61-s + 0.744·65-s + 0.977·67-s − 0.237·71-s − 1.17·73-s + 1.80·79-s − 0.329·83-s − 0.542·85-s + 0.105·89-s − 0.102·95-s + 0.101·97-s + 0.0995·101-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: \(1\)
Selberg data: \((2,\ 97020,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−14.05403931756365, −13.64046196502536, −12.93487133264906, −12.38056715465846, −12.18561983561640, −11.76126914348788, −10.97024511425902, −10.67767400409684, −10.00068507502086, −9.630356818600299, −9.156964069661808, −8.522961172820328, −7.793456125070932, −7.566959460956000, −7.178311695421205, −6.469817758499183, −5.731195604957576, −5.335221745169966, −4.737776792711548, −4.271030777832534, −3.462314388819498, −3.002366602490473, −2.385743007673113, −1.645103665512904, −0.7720663550846192, 0, 0.7720663550846192, 1.645103665512904, 2.385743007673113, 3.002366602490473, 3.462314388819498, 4.271030777832534, 4.737776792711548, 5.335221745169966, 5.731195604957576, 6.469817758499183, 7.178311695421205, 7.566959460956000, 7.793456125070932, 8.522961172820328, 9.156964069661808, 9.630356818600299, 10.00068507502086, 10.67767400409684, 10.97024511425902, 11.76126914348788, 12.18561983561640, 12.38056715465846, 12.93487133264906, 13.64046196502536, 14.05403931756365

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