Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.681 + 0.731i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.271 − 1.38i)2-s − 3-s + (−1.85 + 0.754i)4-s + 0.299i·5-s + (0.271 + 1.38i)6-s − 1.88·7-s + (1.55 + 2.36i)8-s + 9-s + (0.416 − 0.0814i)10-s − 2.47·11-s + (1.85 − 0.754i)12-s − 1.99i·13-s + (0.511 + 2.60i)14-s − 0.299i·15-s + (2.86 − 2.79i)16-s + 0.452·17-s + ⋯
L(s)  = 1  + (−0.192 − 0.981i)2-s − 0.577·3-s + (−0.926 + 0.377i)4-s + 0.134i·5-s + (0.110 + 0.566i)6-s − 0.710·7-s + (0.548 + 0.836i)8-s + 0.333·9-s + (0.131 − 0.0257i)10-s − 0.744·11-s + (0.534 − 0.217i)12-s − 0.554i·13-s + (0.136 + 0.697i)14-s − 0.0774i·15-s + (0.715 − 0.698i)16-s + 0.109·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.681 + 0.731i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.681 + 0.731i)$
$L(1)$  $\approx$  $0.783759 - 0.341012i$
$L(\frac12)$  $\approx$  $0.783759 - 0.341012i$
$L(\frac{3}{2})$   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 \( 1 + (0.271 + 1.38i)T \)
3 \( 1 + T \)
67 \( 1 + (-2.90 - 7.65i)T \)
good5 \( 1 - 0.299iT - 5T^{2} \)
7 \( 1 + 1.88T + 7T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
13 \( 1 + 1.99iT - 13T^{2} \)
17 \( 1 - 0.452T + 17T^{2} \)
19 \( 1 - 5.21iT - 19T^{2} \)
23 \( 1 - 0.807iT - 23T^{2} \)
29 \( 1 - 9.00T + 29T^{2} \)
31 \( 1 - 8.22T + 31T^{2} \)
37 \( 1 - 1.31T + 37T^{2} \)
41 \( 1 + 6.67iT - 41T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 + 2.51iT - 47T^{2} \)
53 \( 1 + 0.990iT - 53T^{2} \)
59 \( 1 + 7.56iT - 59T^{2} \)
61 \( 1 - 9.06iT - 61T^{2} \)
71 \( 1 + 0.0862iT - 71T^{2} \)
73 \( 1 - 1.91T + 73T^{2} \)
79 \( 1 - 8.30T + 79T^{2} \)
83 \( 1 + 0.827iT - 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 - 6.11iT - 97T^{2} \)
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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

−10.32943373538909722333585428206, −9.684014187494583488581721738656, −8.529144455603966060671568390228, −7.79874913021461494490420495091, −6.59800255191303480459350451614, −5.57279042470416873372512488588, −4.65361681016110541434554051178, −3.47030329818192676542172698890, −2.53747671886258372613739419937, −0.852951875550104829337650171678, 0.75672055331039944556574291531, 2.91029007033710530096125756915, 4.50915847272661732419653250056, 5.02090054263504263254343674784, 6.31977029991516718814352079150, 6.63905845223959442734830527149, 7.71155115023407363216829347748, 8.610443753410556879657184006552, 9.468102867119053866799919557516, 10.19815618939584894482870846756

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