Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.997 + 0.0679i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.01 − 0.985i)2-s + 3-s + (0.0573 + 1.99i)4-s + 1.66i·5-s + (−1.01 − 0.985i)6-s + 0.193·7-s + (1.91 − 2.08i)8-s + 9-s + (1.64 − 1.69i)10-s + 5.43·11-s + (0.0573 + 1.99i)12-s − 3.51i·13-s + (−0.196 − 0.190i)14-s + 1.66i·15-s + (−3.99 + 0.229i)16-s − 4.63·17-s + ⋯
L(s)  = 1  + (−0.717 − 0.696i)2-s + 0.577·3-s + (0.0286 + 0.999i)4-s + 0.746i·5-s + (−0.414 − 0.402i)6-s + 0.0731·7-s + (0.676 − 0.736i)8-s + 0.333·9-s + (0.520 − 0.535i)10-s + 1.63·11-s + (0.0165 + 0.577i)12-s − 0.975i·13-s + (−0.0524 − 0.0510i)14-s + 0.430i·15-s + (−0.998 + 0.0573i)16-s − 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.997 + 0.0679i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.997 + 0.0679i)$
$L(1)$  $\approx$  $1.43210 - 0.0487418i$
$L(\frac12)$  $\approx$  $1.43210 - 0.0487418i$
$L(\frac{3}{2})$   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 \( 1 + (1.01 + 0.985i)T \)
3 \( 1 - T \)
67 \( 1 + (-0.790 + 8.14i)T \)
good5 \( 1 - 1.66iT - 5T^{2} \)
7 \( 1 - 0.193T + 7T^{2} \)
11 \( 1 - 5.43T + 11T^{2} \)
13 \( 1 + 3.51iT - 13T^{2} \)
17 \( 1 + 4.63T + 17T^{2} \)
19 \( 1 - 1.14iT - 19T^{2} \)
23 \( 1 - 7.82iT - 23T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 - 8.23T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + 0.746iT - 41T^{2} \)
43 \( 1 - 3.05T + 43T^{2} \)
47 \( 1 + 2.28iT - 47T^{2} \)
53 \( 1 - 6.58iT - 53T^{2} \)
59 \( 1 - 1.92iT - 59T^{2} \)
61 \( 1 - 1.81iT - 61T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 - 5.05T + 73T^{2} \)
79 \( 1 + 8.43T + 79T^{2} \)
83 \( 1 + 8.12iT - 83T^{2} \)
89 \( 1 - 4.29T + 89T^{2} \)
97 \( 1 + 8.73iT - 97T^{2} \)
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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

−10.10155965043259261585594641757, −9.463465681503527456180599674774, −8.735909831090441011677038396433, −7.82235589074452353972522249650, −7.05485162704353878815776201131, −6.16012679675683359111137628724, −4.38177184180353388478021461382, −3.50503100279207821468107597393, −2.60792737120885997419027619552, −1.30603122258834577026268375626, 1.04778954713904512512753500802, 2.27395195931714349247755699363, 4.23810417390396754638382265688, 4.72374130944765833560894411190, 6.42947890215594834075144233208, 6.62544753394453605440559898070, 7.911882049546242872903280230185, 8.777582404644151280470088636624, 9.104211277347421304275664453355, 9.824300732640384237602356267352

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