Properties

Label 2-3-3.2-c162-0-15
Degree $2$
Conductor $3$
Sign $1$
Analytic cond. $498.582$
Root an. cond. $22.3289$
Motivic weight $162$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.43e38·3-s + 5.84e48·4-s − 4.63e68·7-s + 1.96e77·9-s − 2.59e87·12-s − 8.62e88·13-s + 3.41e97·16-s − 7.39e103·19-s + 2.05e107·21-s + 1.71e113·25-s − 8.71e115·27-s − 2.70e117·28-s + 1.24e121·31-s + 1.14e126·36-s − 1.52e127·37-s + 3.82e127·39-s − 3.52e132·43-s − 1.51e136·48-s + 1.34e137·49-s − 5.04e137·52-s + 3.27e142·57-s + 4.73e144·61-s − 9.11e145·63-s + 1.99e146·64-s − 1.63e148·67-s − 1.25e151·73-s − 7.58e151·75-s + ⋯
L(s)  = 1  − 3-s + 4-s − 1.63·7-s + 9-s − 12-s − 0.0508·13-s + 16-s − 1.94·19-s + 1.63·21-s + 25-s − 27-s − 1.63·28-s + 1.97·31-s + 36-s − 1.44·37-s + 0.0508·39-s − 1.72·43-s − 48-s + 1.66·49-s − 0.0508·52-s + 1.94·57-s + 1.15·61-s − 1.63·63-s + 64-s − 1.99·67-s − 1.47·73-s − 75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(163-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+81) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Analytic conductor: \(498.582\)
Root analytic conductor: \(22.3289\)
Motivic weight: \(162\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :81),\ 1)\)

Particular Values

\(L(\frac{163}{2})\) \(\approx\) \(0.8891374927\)
\(L(\frac12)\) \(\approx\) \(0.8891374927\)
\(L(82)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{81} T \)
good2 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
5 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
7 \( 1 + \)\(46\!\cdots\!14\)\( T + p^{162} T^{2} \)
11 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
13 \( 1 + \)\(86\!\cdots\!26\)\( T + p^{162} T^{2} \)
17 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
19 \( 1 + \)\(73\!\cdots\!62\)\( T + p^{162} T^{2} \)
23 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
29 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
31 \( 1 - \)\(12\!\cdots\!62\)\( T + p^{162} T^{2} \)
37 \( 1 + \)\(15\!\cdots\!74\)\( T + p^{162} T^{2} \)
41 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
43 \( 1 + \)\(35\!\cdots\!86\)\( T + p^{162} T^{2} \)
47 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
53 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
59 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
61 \( 1 - \)\(47\!\cdots\!22\)\( T + p^{162} T^{2} \)
67 \( 1 + \)\(16\!\cdots\!34\)\( T + p^{162} T^{2} \)
71 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
73 \( 1 + \)\(12\!\cdots\!46\)\( T + p^{162} T^{2} \)
79 \( 1 + \)\(60\!\cdots\!42\)\( T + p^{162} T^{2} \)
83 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
89 \( ( 1 - p^{81} T )( 1 + p^{81} T ) \)
97 \( 1 - \)\(12\!\cdots\!06\)\( T + p^{162} T^{2} \)
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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

−10.11783999944372795033525005120, −8.593167337047669282123797531781, −7.04069697896008390529952709666, −6.53756309319070024788768014086, −5.93903918882746854997198435093, −4.62876655275776987041328147602, −3.45523036334612448573078309614, −2.54473923250732832187444871326, −1.44714423338764113914967176894, −0.34909254111996443773051951042, 0.34909254111996443773051951042, 1.44714423338764113914967176894, 2.54473923250732832187444871326, 3.45523036334612448573078309614, 4.62876655275776987041328147602, 5.93903918882746854997198435093, 6.53756309319070024788768014086, 7.04069697896008390529952709666, 8.593167337047669282123797531781, 10.11783999944372795033525005120

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