Properties

Degree 2
Conductor 3
Sign $1$
Motivic weight 37
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.10e4·2-s + 3.87e8·3-s − 1.30e11·4-s − 7.16e12·5-s − 3.14e13·6-s − 5.96e15·7-s + 2.17e16·8-s + 1.50e17·9-s + 5.80e17·10-s + 2.06e19·11-s − 5.07e19·12-s − 4.02e20·13-s + 4.83e20·14-s − 2.77e21·15-s + 1.62e22·16-s − 4.55e22·17-s − 1.21e22·18-s + 1.54e23·19-s + 9.37e23·20-s − 2.31e24·21-s − 1.67e24·22-s + 2.42e25·23-s + 8.42e24·24-s − 2.14e25·25-s + 3.26e25·26-s + 5.81e25·27-s + 7.81e26·28-s + ⋯
L(s)  = 1  − 0.218·2-s + 0.577·3-s − 0.952·4-s − 0.840·5-s − 0.126·6-s − 1.38·7-s + 0.426·8-s + 0.333·9-s + 0.183·10-s + 1.11·11-s − 0.549·12-s − 0.993·13-s + 0.302·14-s − 0.485·15-s + 0.858·16-s − 0.786·17-s − 0.0728·18-s + 0.341·19-s + 0.799·20-s − 0.799·21-s − 0.244·22-s + 1.55·23-s + 0.246·24-s − 0.294·25-s + 0.217·26-s + 0.192·27-s + 1.31·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(37\)
character  :  $\chi_{3} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3,\ (\ :37/2),\ 1)$
$L(19)$  $\approx$  $1.019761564$
$L(\frac12)$  $\approx$  $1.019761564$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 3$, \(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - 3.87e8T \)
good2 \( 1 + 8.10e4T + 1.37e11T^{2} \)
5 \( 1 + 7.16e12T + 7.27e25T^{2} \)
7 \( 1 + 5.96e15T + 1.85e31T^{2} \)
11 \( 1 - 2.06e19T + 3.40e38T^{2} \)
13 \( 1 + 4.02e20T + 1.64e41T^{2} \)
17 \( 1 + 4.55e22T + 3.36e45T^{2} \)
19 \( 1 - 1.54e23T + 2.06e47T^{2} \)
23 \( 1 - 2.42e25T + 2.42e50T^{2} \)
29 \( 1 - 1.52e27T + 1.28e54T^{2} \)
31 \( 1 - 6.09e27T + 1.51e55T^{2} \)
37 \( 1 + 6.28e28T + 1.05e58T^{2} \)
41 \( 1 + 8.11e29T + 4.70e59T^{2} \)
43 \( 1 - 4.69e29T + 2.74e60T^{2} \)
47 \( 1 - 1.02e31T + 7.37e61T^{2} \)
53 \( 1 - 5.42e31T + 6.28e63T^{2} \)
59 \( 1 + 5.27e32T + 3.32e65T^{2} \)
61 \( 1 + 6.19e31T + 1.14e66T^{2} \)
67 \( 1 - 9.30e33T + 3.67e67T^{2} \)
71 \( 1 + 2.75e34T + 3.13e68T^{2} \)
73 \( 1 - 2.09e34T + 8.76e68T^{2} \)
79 \( 1 - 1.20e34T + 1.63e70T^{2} \)
83 \( 1 + 3.25e34T + 1.01e71T^{2} \)
89 \( 1 + 1.39e36T + 1.34e72T^{2} \)
97 \( 1 + 6.18e36T + 3.24e73T^{2} \)
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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

−17.11956316134773396463673097948, −15.40714510733639842267820407196, −13.75268880621634290443403524157, −12.28962476896634146619763410342, −9.831991251053794891370908828608, −8.712982752843244742574982245931, −6.94441566309081609996071473436, −4.42275179296483521805118028822, −3.14887007459806955539309380437, −0.65964473813392773060173476615, 0.65964473813392773060173476615, 3.14887007459806955539309380437, 4.42275179296483521805118028822, 6.94441566309081609996071473436, 8.712982752843244742574982245931, 9.831991251053794891370908828608, 12.28962476896634146619763410342, 13.75268880621634290443403524157, 15.40714510733639842267820407196, 17.11956316134773396463673097948

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