Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.689 + 0.724i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.946 + 1.05i)2-s − 3-s + (−0.208 − 1.98i)4-s + 4.42i·5-s + (0.946 − 1.05i)6-s − 2.81·7-s + (2.28 + 1.66i)8-s + 9-s + (−4.65 − 4.19i)10-s − 4.49·11-s + (0.208 + 1.98i)12-s − 6.44i·13-s + (2.66 − 2.96i)14-s − 4.42i·15-s + (−3.91 + 0.831i)16-s + 1.33·17-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s − 0.577·3-s + (−0.104 − 0.994i)4-s + 1.98i·5-s + (0.386 − 0.429i)6-s − 1.06·7-s + (0.808 + 0.587i)8-s + 0.333·9-s + (−1.47 − 1.32i)10-s − 1.35·11-s + (0.0603 + 0.574i)12-s − 1.78i·13-s + (0.712 − 0.791i)14-s − 1.14i·15-s + (−0.978 + 0.207i)16-s + 0.323·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.689 + 0.724i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.689 + 0.724i)$
$L(1)$  $\approx$  $0.176644 - 0.0756862i$
$L(\frac12)$  $\approx$  $0.176644 - 0.0756862i$
$L(\frac{3}{2})$   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 \( 1 + (0.946 - 1.05i)T \)
3 \( 1 + T \)
67 \( 1 + (-6.48 + 4.99i)T \)
good5 \( 1 - 4.42iT - 5T^{2} \)
7 \( 1 + 2.81T + 7T^{2} \)
11 \( 1 + 4.49T + 11T^{2} \)
13 \( 1 + 6.44iT - 13T^{2} \)
17 \( 1 - 1.33T + 17T^{2} \)
19 \( 1 - 1.70iT - 19T^{2} \)
23 \( 1 - 3.41iT - 23T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 - 8.45T + 31T^{2} \)
37 \( 1 - 5.75T + 37T^{2} \)
41 \( 1 + 5.15iT - 41T^{2} \)
43 \( 1 - 1.06T + 43T^{2} \)
47 \( 1 - 4.52iT - 47T^{2} \)
53 \( 1 + 4.63iT - 53T^{2} \)
59 \( 1 + 10.2iT - 59T^{2} \)
61 \( 1 + 9.66iT - 61T^{2} \)
71 \( 1 + 0.479iT - 71T^{2} \)
73 \( 1 - 3.68T + 73T^{2} \)
79 \( 1 + 9.91T + 79T^{2} \)
83 \( 1 + 8.53iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−10.00922994834936293491427087571, −9.829474592030711467780890344806, −7.933543169346593991021564063295, −7.64885109435573097390766150362, −6.63307346365207904534425067764, −6.01407626706708947938525488874, −5.31927259386113053121799164769, −3.43767873522875051296766721393, −2.56405782189240401926755215632, −0.14761263995010301939642637944, 1.09902493018312668013453133400, 2.50206474486116538286834346635, 4.16390882135659158558732927822, 4.72385082760471891592877341821, 5.89972852377914666146855084662, 7.06568145696874789917844314317, 8.145433746361898541041839913727, 8.851308086067874756412946802845, 9.614225033041135187178995477548, 10.08978344625643043663147174281

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