Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $-0.986 - 0.165i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.219 + 1.39i)2-s − 3-s + (−1.90 − 0.613i)4-s + 3.15i·5-s + (0.219 − 1.39i)6-s + 0.761·7-s + (1.27 − 2.52i)8-s + 9-s + (−4.41 − 0.693i)10-s + 5.14·11-s + (1.90 + 0.613i)12-s + 4.67i·13-s + (−0.167 + 1.06i)14-s − 3.15i·15-s + (3.24 + 2.33i)16-s + 1.78·17-s + ⋯
L(s)  = 1  + (−0.155 + 0.987i)2-s − 0.577·3-s + (−0.951 − 0.306i)4-s + 1.41i·5-s + (0.0895 − 0.570i)6-s + 0.287·7-s + (0.450 − 0.892i)8-s + 0.333·9-s + (−1.39 − 0.219i)10-s + 1.55·11-s + (0.549 + 0.177i)12-s + 1.29i·13-s + (−0.0446 + 0.284i)14-s − 0.815i·15-s + (0.811 + 0.583i)16-s + 0.433·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $-0.986 - 0.165i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (535, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ -0.986 - 0.165i)$
$L(1)$  $\approx$  $0.0890661 + 1.06598i$
$L(\frac12)$  $\approx$  $0.0890661 + 1.06598i$
$L(\frac{3}{2})$   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 \( 1 + (0.219 - 1.39i)T \)
3 \( 1 + T \)
67 \( 1 + (8.09 - 1.18i)T \)
good5 \( 1 - 3.15iT - 5T^{2} \)
7 \( 1 - 0.761T + 7T^{2} \)
11 \( 1 - 5.14T + 11T^{2} \)
13 \( 1 - 4.67iT - 13T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 - 5.49iT - 19T^{2} \)
23 \( 1 + 2.96iT - 23T^{2} \)
29 \( 1 + 9.79T + 29T^{2} \)
31 \( 1 - 3.67T + 31T^{2} \)
37 \( 1 - 8.38T + 37T^{2} \)
41 \( 1 + 1.00iT - 41T^{2} \)
43 \( 1 + 7.22T + 43T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 - 9.25iT - 53T^{2} \)
59 \( 1 + 0.539iT - 59T^{2} \)
61 \( 1 - 2.35iT - 61T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + 7.50T + 73T^{2} \)
79 \( 1 - 4.41T + 79T^{2} \)
83 \( 1 + 8.49iT - 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 8.80iT - 97T^{2} \)
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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

−10.51321635835905622548487438795, −9.770271739863485301990477657215, −8.988485762645386960729969863562, −7.82492396023955790493215714876, −6.97423231856655867616122763925, −6.45260269774812791028912202588, −5.77104045488538627745506469921, −4.36433426182299770822298961838, −3.64356924435445570047891022051, −1.63811198483392994699949593478, 0.69019199637017262044515562250, 1.58320172633497011379782685225, 3.35194145865587026697262215923, 4.46011661258106756615704470450, 5.09204442197960892270375689236, 6.04419723094633693950005862553, 7.57267058795044540841495401347, 8.384525753861432769584243545910, 9.329658457793976135684378714714, 9.651379138376102131990111545171

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