Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.140 - 0.990i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.16 + 0.799i)2-s − 3-s + (0.720 + 1.86i)4-s + 0.864i·5-s + (−1.16 − 0.799i)6-s + 1.73·7-s + (−0.651 + 2.75i)8-s + 9-s + (−0.691 + 1.00i)10-s + 5.29·11-s + (−0.720 − 1.86i)12-s − 1.41i·13-s + (2.02 + 1.38i)14-s − 0.864i·15-s + (−2.96 + 2.68i)16-s − 4.82·17-s + ⋯
L(s)  = 1  + (0.824 + 0.565i)2-s − 0.577·3-s + (0.360 + 0.932i)4-s + 0.386i·5-s + (−0.476 − 0.326i)6-s + 0.655·7-s + (−0.230 + 0.973i)8-s + 0.333·9-s + (−0.218 + 0.318i)10-s + 1.59·11-s + (−0.207 − 0.538i)12-s − 0.391i·13-s + (0.540 + 0.370i)14-s − 0.223i·15-s + (−0.740 + 0.672i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.140 - 0.990i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.140 - 0.990i)$
$L(1)$  $\approx$  $1.48344 + 1.70877i$
$L(\frac12)$  $\approx$  $1.48344 + 1.70877i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;67\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.799i)T \)
3 \( 1 + T \)
67 \( 1 + (-7.97 - 1.84i)T \)
good5 \( 1 - 0.864iT - 5T^{2} \)
7 \( 1 - 1.73T + 7T^{2} \)
11 \( 1 - 5.29T + 11T^{2} \)
13 \( 1 + 1.41iT - 13T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 - 4.26iT - 19T^{2} \)
23 \( 1 - 1.92iT - 23T^{2} \)
29 \( 1 + 2.13T + 29T^{2} \)
31 \( 1 - 5.52T + 31T^{2} \)
37 \( 1 + 7.24T + 37T^{2} \)
41 \( 1 - 0.306iT - 41T^{2} \)
43 \( 1 + 1.13T + 43T^{2} \)
47 \( 1 - 11.9iT - 47T^{2} \)
53 \( 1 - 8.91iT - 53T^{2} \)
59 \( 1 + 14.4iT - 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 8.20T + 79T^{2} \)
83 \( 1 - 0.390iT - 83T^{2} \)
89 \( 1 + 5.73T + 89T^{2} \)
97 \( 1 + 14.5iT - 97T^{2} \)
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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

−10.97098927610772630973593582873, −9.601135683231801959841150860126, −8.588709189087377204195953911870, −7.72271831623207778428937351015, −6.66246127488580625129864246392, −6.29160334635905830408023808367, −5.12501960089707910534259642686, −4.31613365926688369647743772239, −3.34828660525836886888802270379, −1.74813041968215313444118646087, 1.03113108402797166826363700924, 2.20263302654570733359063539438, 3.83002860709773391866718525971, 4.59494355108010983052381519442, 5.28425363114198328641599395851, 6.62949119712470806612215864464, 6.83530942688501912479376950456, 8.628161547252042375156803885523, 9.227832194934497443194578985064, 10.31081904196380194845766387339

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