Properties

Degree 2
Conductor 3
Sign $1$
Motivic weight 42
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.04e10·3-s + 4.39e12·4-s + 1.46e17·7-s + 1.09e20·9-s − 4.60e22·12-s − 3.57e23·13-s + 1.93e25·16-s − 1.65e26·19-s − 1.52e27·21-s + 2.27e29·25-s − 1.14e30·27-s + 6.43e29·28-s + 4.00e31·31-s + 4.81e32·36-s + 1.62e33·37-s + 3.73e33·39-s + 3.01e34·43-s − 2.02e35·48-s − 2.90e35·49-s − 1.57e36·52-s + 1.72e36·57-s + 5.58e37·61-s + 1.60e37·63-s + 8.50e37·64-s − 3.97e38·67-s − 1.16e39·73-s − 2.37e39·75-s + ⋯
L(s)  = 1  − 3-s + 4-s + 0.261·7-s + 9-s − 12-s − 1.44·13-s + 16-s − 0.231·19-s − 0.261·21-s + 25-s − 27-s + 0.261·28-s + 1.92·31-s + 36-s + 1.90·37-s + 1.44·39-s + 1.50·43-s − 48-s − 0.931·49-s − 1.44·52-s + 0.231·57-s + 1.80·61-s + 0.261·63-s + 64-s − 1.78·67-s − 0.866·73-s − 75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(43-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(42\)
character  :  $\chi_{3} (2, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3,\ (\ :21),\ 1)$
$L(\frac{43}{2})$  $\approx$  $1.846569382$
$L(\frac12)$  $\approx$  $1.846569382$
$L(22)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p\) is a polynomial of degree 2. If $p = 3$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + p^{21} T \)
good2 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
5 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
7 \( 1 - 146246101081752386 T + p^{42} T^{2} \)
11 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
13 \( 1 + \)\(35\!\cdots\!26\)\( T + p^{42} T^{2} \)
17 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
19 \( 1 + \)\(16\!\cdots\!62\)\( T + p^{42} T^{2} \)
23 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
29 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
31 \( 1 - \)\(40\!\cdots\!62\)\( T + p^{42} T^{2} \)
37 \( 1 - \)\(16\!\cdots\!26\)\( T + p^{42} T^{2} \)
41 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
43 \( 1 - \)\(30\!\cdots\!14\)\( T + p^{42} T^{2} \)
47 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
53 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
59 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
61 \( 1 - \)\(55\!\cdots\!22\)\( T + p^{42} T^{2} \)
67 \( 1 + \)\(39\!\cdots\!34\)\( T + p^{42} T^{2} \)
71 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
73 \( 1 + \)\(11\!\cdots\!46\)\( T + p^{42} T^{2} \)
79 \( 1 - \)\(14\!\cdots\!58\)\( T + p^{42} T^{2} \)
83 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
89 \( ( 1 - p^{21} T )( 1 + p^{21} T ) \)
97 \( 1 - \)\(22\!\cdots\!06\)\( T + p^{42} T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.35827235700219296252281937246, −14.91787482411171572568475666686, −12.51198673522446325747556593263, −11.39928953835505891389508257875, −10.08796942496801273191380363359, −7.51929882772980564874026072286, −6.25969698317905202930938137396, −4.72969493735444101595200033906, −2.48208481356831408569546049547, −0.890144053637415040605839661333, 0.890144053637415040605839661333, 2.48208481356831408569546049547, 4.72969493735444101595200033906, 6.25969698317905202930938137396, 7.51929882772980564874026072286, 10.08796942496801273191380363359, 11.39928953835505891389508257875, 12.51198673522446325747556593263, 14.91787482411171572568475666686, 16.35827235700219296252281937246

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