Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.992 - 0.124i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.58 − 0.707i)3-s + 3.16·5-s + 1.41i·7-s + (2.00 + 2.23i)9-s + 3.16·11-s − 4.24i·13-s + (−5.00 − 2.23i)15-s + 6.70i·17-s − 3·19-s + (1.00 − 2.23i)21-s + 6.70i·23-s + 5.00·25-s + (−1.58 − 4.94i)27-s − 2.23i·29-s + 4.24i·31-s + ⋯
L(s)  = 1  + (−0.912 − 0.408i)3-s + 1.41·5-s + 0.534i·7-s + (0.666 + 0.745i)9-s + 0.953·11-s − 1.17i·13-s + (−1.29 − 0.577i)15-s + 1.62i·17-s − 0.688·19-s + (0.218 − 0.487i)21-s + 1.39i·23-s + 1.00·25-s + (−0.304 − 0.952i)27-s − 0.415i·29-s + 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.992 - 0.124i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (401, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.992 - 0.124i)$
$L(1)$  $\approx$  $1.54881 + 0.0964237i$
$L(\frac12)$  $\approx$  $1.54881 + 0.0964237i$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 + (1.58 + 0.707i)T \)
67 \( 1 + (7 - 4.24i)T \)
good5 \( 1 - 3.16T + 5T^{2} \)
7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 - 3.16T + 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 - 6.70iT - 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 - 6.70iT - 23T^{2} \)
29 \( 1 + 2.23iT - 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 - 12.6T + 41T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 + 2.23iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 2.23iT - 59T^{2} \)
61 \( 1 - 2.82iT - 61T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 - 4.47iT - 83T^{2} \)
89 \( 1 - 2.23iT - 89T^{2} \)
97 \( 1 + 8.48iT - 97T^{2} \)
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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

−10.37725068281455697535919945455, −9.567324188810479551833918946478, −8.669382773459439844681958704413, −7.60107474010325987211904997078, −6.43563795276384791839368163809, −5.88143900753117517513427491639, −5.37945793692083760899248127196, −3.95467863585309783018298840209, −2.29009983163835698543836546844, −1.31851204504182609025724914495, 1.05232544094010160938320677403, 2.45590510625300637903520791201, 4.20014960372126109493567815418, 4.77892523546812327037685112636, 6.07826747486617672411190416660, 6.43960915658412676994465965770, 7.34822885380022822483018009780, 9.105108704153527324203743589833, 9.390089349175925021436380624132, 10.19414330895103725277636173339

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