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Properties

Label 2-270802-1.1-c1-0-19
Degree $2$
Conductor $270802$
Sign $-1$
Analytic cond. $2162.36$
Root an. cond. $46.5012$
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  + 2-s − 3·3-s + 4-s + 3·5-s − 3·6-s − 7-s + 8-s + 6·9-s + 3·10-s + 5·11-s − 3·12-s + 13-s − 14-s − 9·15-s + 16-s + 2·17-s + 6·18-s − 4·19-s + 3·20-s + 3·21-s + 5·22-s − 23-s − 3·24-s + 4·25-s + 26-s − 9·27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.34·5-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s + 0.948·10-s + 1.50·11-s − 0.866·12-s + 0.277·13-s − 0.267·14-s − 2.32·15-s + 1/4·16-s + 0.485·17-s + 1.41·18-s − 0.917·19-s + 0.670·20-s + 0.654·21-s + 1.06·22-s − 0.208·23-s − 0.612·24-s + 4/5·25-s + 0.196·26-s − 1.73·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(270802\)    =    \(2 \cdot 7 \cdot 23 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(2162.36\)
Root analytic conductor: \(46.5012\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 270802,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−12.99049391122675, −12.53252144767701, −11.98894733209816, −11.77266857934965, −11.26729364920148, −10.85939029473229, −10.14225976561037, −9.965257451701184, −9.730471468496832, −8.816326067487483, −8.519069157102211, −7.634485285429744, −6.864739045672745, −6.583431886797724, −6.368003358620568, −5.954053575718649, −5.454752613283802, −4.988959586130437, −4.550613458508954, −3.903596473037117, −3.474327692991027, −2.640074397492777, −1.847402359506438, −1.481566618009072, −0.9287327514051933, 0, 0.9287327514051933, 1.481566618009072, 1.847402359506438, 2.640074397492777, 3.474327692991027, 3.903596473037117, 4.550613458508954, 4.988959586130437, 5.454752613283802, 5.954053575718649, 6.368003358620568, 6.583431886797724, 6.864739045672745, 7.634485285429744, 8.519069157102211, 8.816326067487483, 9.730471468496832, 9.965257451701184, 10.14225976561037, 10.85939029473229, 11.26729364920148, 11.77266857934965, 11.98894733209816, 12.53252144767701, 12.99049391122675

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