Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 67 $
Sign $0.967 - 0.253i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 − 1.13i)3-s + (1.65 + 4.13i)7-s + (0.426 − 2.96i)9-s + (−0.540 + 5.66i)13-s + (7.29 + 2.92i)19-s + (6.85 + 3.53i)21-s + (−2.07 − 4.54i)25-s + (−2.80 − 4.37i)27-s + (0.426 + 4.46i)31-s + (1.07 − 1.86i)37-s + (5.71 + 8.02i)39-s + (−3.53 − 12.0i)43-s + (−9.27 + 8.83i)49-s + (12.8 − 4.45i)57-s + (10.9 − 0.521i)61-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)3-s + (0.625 + 1.56i)7-s + (0.142 − 0.989i)9-s + (−0.149 + 1.57i)13-s + (1.67 + 0.669i)19-s + (1.49 + 0.770i)21-s + (−0.415 − 0.909i)25-s + (−0.540 − 0.841i)27-s + (0.0766 + 0.802i)31-s + (0.176 − 0.305i)37-s + (0.915 + 1.28i)39-s + (−0.538 − 1.83i)43-s + (−1.32 + 1.26i)49-s + (1.70 − 0.589i)57-s + (1.40 − 0.0667i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
\( \varepsilon \)  =  $0.967 - 0.253i$
motivic weight  =  \(1\)
character  :  $\chi_{804} (701, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 804,\ (\ :1/2),\ 0.967 - 0.253i)$
$L(1)$  $\approx$  $2.11889 + 0.272645i$
$L(\frac12)$  $\approx$  $2.11889 + 0.272645i$
$L(\frac{3}{2})$   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 \( 1 + (-1.30 + 1.13i)T \)
67 \( 1 + (-3.22 + 7.52i)T \)
good5 \( 1 + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (-1.65 - 4.13i)T + (-5.06 + 4.83i)T^{2} \)
11 \( 1 + (-10.9 - 1.04i)T^{2} \)
13 \( 1 + (0.540 - 5.66i)T + (-12.7 - 2.46i)T^{2} \)
17 \( 1 + (15.1 + 7.78i)T^{2} \)
19 \( 1 + (-7.29 - 2.92i)T + (13.7 + 13.1i)T^{2} \)
23 \( 1 + (-21.3 + 8.54i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.426 - 4.46i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (-1.07 + 1.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.95 + 40.9i)T^{2} \)
43 \( 1 + (3.53 + 12.0i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (36.9 + 29.0i)T^{2} \)
53 \( 1 + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (38.6 + 44.5i)T^{2} \)
61 \( 1 + (-10.9 + 0.521i)T + (60.7 - 5.79i)T^{2} \)
71 \( 1 + (63.1 - 32.5i)T^{2} \)
73 \( 1 + (0.120 + 2.53i)T + (-72.6 + 6.93i)T^{2} \)
79 \( 1 + (13.7 + 9.78i)T + (25.8 + 74.6i)T^{2} \)
83 \( 1 + (-48.1 + 67.6i)T^{2} \)
89 \( 1 + (12.6 + 88.0i)T^{2} \)
97 \( 1 + (-14.0 - 8.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.02652940962667870660061161733, −9.169407265406759866959135061668, −8.683267011054571573442254846379, −7.83602555104292202823506412026, −6.92630397510437350625599810402, −5.97234792371543585141849253271, −5.00659961520100355707788993712, −3.68062227207101510105836851440, −2.45092345822915643489807581411, −1.65614640634787770831844324184, 1.11918731349471667907563301902, 2.88042265295991558458125107098, 3.72746651005964986574268899181, 4.72729047201633702802790931993, 5.51254829096677499508333469181, 7.18791089642849042166772063044, 7.70073797088671080994444765313, 8.368823599495076524744565382535, 9.710889142305277976526290040672, 9.988307254969628040756340585071

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