Properties

Degree $2$
Conductor $3$
Sign $1$
Motivic weight $66$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.55e15·3-s + 7.37e19·4-s − 1.54e28·7-s + 3.09e31·9-s − 4.10e35·12-s − 1.09e37·13-s + 5.44e39·16-s − 8.32e41·19-s + 8.59e43·21-s + 1.35e46·25-s − 1.71e47·27-s − 1.14e48·28-s − 2.08e49·31-s + 2.28e51·36-s − 7.41e51·37-s + 6.11e52·39-s − 1.24e54·43-s − 3.02e55·48-s + 1.79e56·49-s − 8.11e56·52-s + 4.62e57·57-s + 7.67e58·61-s − 4.77e59·63-s + 4.01e59·64-s + 2.36e59·67-s + 5.64e61·73-s − 7.53e61·75-s + ⋯
L(s)  = 1  − 3-s + 4-s − 1.99·7-s + 9-s − 12-s − 1.90·13-s + 16-s − 0.526·19-s + 1.99·21-s + 25-s − 27-s − 1.99·28-s − 1.26·31-s + 36-s − 1.31·37-s + 1.90·39-s − 1.55·43-s − 48-s + 2.99·49-s − 1.90·52-s + 0.526·57-s + 0.931·61-s − 1.99·63-s + 64-s + 0.129·67-s + 1.82·73-s − 75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Motivic weight: \(66\)
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :33),\ 1)\)

Particular Values

\(L(\frac{67}{2})\) \(\approx\) \(0.7517750513\)
\(L(\frac12)\) \(\approx\) \(0.7517750513\)
\(L(34)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{33} T \)
good2 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
5 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
7 \( 1 + \)\(15\!\cdots\!14\)\( T + p^{66} T^{2} \)
11 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
13 \( 1 + \)\(10\!\cdots\!06\)\( T + p^{66} T^{2} \)
17 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
19 \( 1 + \)\(83\!\cdots\!82\)\( T + p^{66} T^{2} \)
23 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
29 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
31 \( 1 + \)\(20\!\cdots\!18\)\( T + p^{66} T^{2} \)
37 \( 1 + \)\(74\!\cdots\!94\)\( T + p^{66} T^{2} \)
41 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
43 \( 1 + \)\(12\!\cdots\!86\)\( T + p^{66} T^{2} \)
47 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
53 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
59 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
61 \( 1 - \)\(76\!\cdots\!62\)\( T + p^{66} T^{2} \)
67 \( 1 - \)\(23\!\cdots\!26\)\( T + p^{66} T^{2} \)
71 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
73 \( 1 - \)\(56\!\cdots\!34\)\( T + p^{66} T^{2} \)
79 \( 1 - \)\(62\!\cdots\!78\)\( T + p^{66} T^{2} \)
83 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
89 \( ( 1 - p^{33} T )( 1 + p^{33} T ) \)
97 \( 1 - \)\(24\!\cdots\!46\)\( T + p^{66} T^{2} \)
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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

−12.78407403201732147765382062646, −12.14051631531251486102675075432, −10.55089313114295350873809739171, −9.660027127175615603053669934471, −7.12022491189933960705940185014, −6.57641598385532328528428809723, −5.27051716898601141469677493711, −3.42730664915912797563770001873, −2.17971326698148545590358494805, −0.42779316396945469062926183113, 0.42779316396945469062926183113, 2.17971326698148545590358494805, 3.42730664915912797563770001873, 5.27051716898601141469677493711, 6.57641598385532328528428809723, 7.12022491189933960705940185014, 9.660027127175615603053669934471, 10.55089313114295350873809739171, 12.14051631531251486102675075432, 12.78407403201732147765382062646

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